Hey everyone here's my thoughts on the cards.
I'm creating the following terminology when talking about different types of extra turn cards
Level 0-
Once per game. Cards in this category can gain one extra turn the entire game. Fleet is in this category.
Level 1-
Once per turn on one turn (per round). Cards in this category can be played only once per turn, and only allow one extra turn per round (usually because they don't allow extra turns on any type of extra turn). Outpost and Mission would fall in this category.
Level 2-
Once per turn over multiple turns. Cards in this category can only be played once per turn, but can be played on extra turns (like outpost and mission turns) that are not its own.
Level 3-
Multiple per turn on one turn. Cards in this category can be played multiple times on a turn, but can't be utilized on extra turns (its own or others).
Level 4-
Multiple per turn over multiple turns. Cards in this category can be played multiple times on a single turn, and over multiple turns per round (though not on its own extra turns, or else infinite loops would be possible). Possession would fall in this category.
Level 5-
Infinite Loop. An extra turn card that could allow for infinite turns (usually because it allows itself to be played on its own extra turns).
Other-
Cards that don't fall in these categories.
Double Shift by Pst
Double Shift- Project - $5
After one of your turns in which you gain a Victory card costing more than $5 you get an extra turn. During that extra turn, whenever you would gain a Victory card, instead gain nothing.
A level 2 extra turn. An interesting concept. It's going to be used toward the end, once you've started greening, but you can't use the extra turn to get more points (normally; edge cases always apply). Instead, you'll use it to keep your deck composition. Assuming there aren't nobles, harem, or farmland in the game it will just trigger on provinces meaning you'll probably get ~3-4 extra turns on it. If there's a good enough engine to double-province, you'll get even less. I think it would be better as a level 1, but other than that I like it.
Night Shift by Aquila
Night Shift - Event - $2
Once per game: take another turn after this one, starting with 0 Actions.
A level 0 extra turn. I'm not against level 0 extra turns, I just think this was a bad case to use it. I like the concept of allowing an extra turn with no actions. It's something I might even be able to build a deck around to take advantage of that extra turn (just like one does with outpost). But if I can only get 1 extra turn the whole game, it's not worth building my deck around it, so it makes this card much less interesting. I think this might work better as an action card (I'm guessing the only reason it was an event was to make it once per game) and changed to a level 1 extra turn (you'll probably have to tweak the price).
Pact by Grrgrrgrr
Pact - Project - $8
For the rest of the game, play 2 turns in a row. When you trash a Copper, move it to the discard.
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After buying this project, gain 7 Coppers.
A permanent Level 1 extra turn at the cost of having (at least) 7 untrashable coppers. I like this idea. The fact that you get an extra turn every round isn't as crazy as it first appeared since a well constructed engine can do the same with a single buy of outpost. I think this compares well to outpost in general. Outpost gives an extra turn with only 3 cards. With Pact, you'll have so many coppers, it'll be like having less cards. If you had 21 "good" cards and 7 coppers, you can expect 1.25 cards in each hand to be a copper making it harder to run your engine. The downside is this hits you even on your non-extra turns. In engines, trashing is a must for this buy (14 coppers is just too much).This could be used more effectively for extra turns by BM (than other extra turn cards) since BM usually isn't drawing your deck every turn. In addition BM doesn't care as much about copper. Trashing would be helpful for BM. Alt VP (especially gardens) could probably benefit from this without trashing at all, since your price points are usually lower. Of course $8 is hard to reach for alt VP usually.
Despite all of my analysis, this is one of those cards I think I would understand better with playtesting. There might need to be some tweaking (maybe gain 6 coppers or 8 coppers). I think the price could definitely use some tweaking. It seems like a good fit for BM and Alt VP yet for BM it doesn't work because you have give up a province for it (BM advantage is usually speed), and Alt VP usually has trouble hitting $8. Maybe lower the price or convert it to debt?
In addition, trader can still work here. Maybe instead of gaining the coppers, you just put them in your discard?
Bonanza by mandiaco15Bonanza - Event - $5
At the end of this turn, if you have unspent coins, take Voyager, and take another turn after this one, where you draw one card per unspent coin for your hand.
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You can't buy this if you have Voyager.
Voyager - State
Worth 1VP if you have this at the end of the game.
I think the intention was a Level 1 extra turn but because you take Voyager at the end of the turn this is technically a Level 3. The simplest fix is either a "once per turn" clause or taking voyager when you buy it. That aside, I don't see this being used many times in a game. In order to be a worthwhile buy, you probably need to be producing a large amount of money each turn, and buy something else with this. If I have $10 and buy this and leave the $5 for 5 cards, I would need to be confident I'm going to get more than $10 on this extra turn to have gained anything(why didn't that happen this turn?). Buying this at $7 (2 cards for next turn) seems like a long shot, and at $8 or more (outpost level), I could've just gotten a province. A likely scenario is I have $18 and 2 (or more) buys. Buy province and Bonanza (leave 5 cards for next turn) and then double province next turn. At that point, I have to wait for my oponent to have a worthwhile turn to buy bonanza, and at that point the games almost over. I might not even be able to buy bonanza a 2nd time. I'm sure there are exceptions (guide, hireling, etc.) but I think this is the likely case. I think the choices on this aren't so intriguing. I liked the concept of Voyager (by taking an extra turn, you allow your opponent to take an extra turn. likewise, you can deny your opponent extra turns by not taking extra turns yourself), but when it's expected to pass back and forth so few times, it's not as interesting. I also think Voyager would be a little more interesting as a penalty (-1VP) since whoever takes the last extra turn, is going to have a slight advantage.
Coffee Roast by Fragasnap
Coffee Roast - Treasure - $4P
$2
+2 Cards
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When you buy this, you may trash a Potion you have in play. If you do, take another turn after this one.
A level 5 extra turn. Not only is the card itself pretty strong, 2 peddlers and a lab (even though it's during you buy phase), but it gives an extra turn on top of this (at the cost of a potion). If this were limited to one per round, this could be an ok card. I see this working better in BM deck since it reduces the chance of terminal collision with your actions. It could likely be better than a gold (sometimes even a platinum) if it draws the right cards. I think the glaring error is the infinite turn possibility with trader and treasurer. Any infinite turn loop can be abused (add monument, goons, anything that gains coffers, etc.). The fact that this only requires 3 cards is a problem (see KC-Goons-Masquerade).
Refuge by Kudasai
Refuge - Night - $8
Name a phase.
If this is the first time you played a Refuge this turn, take an extra turn after this one, where you can only play your Clean-up and phases named this turn with Refuge.
A Level 5 extra turn. I know you acknowledged the fact that this can infinite turn loop, but it can really be abused even with only 2 with werewolf. A deck of 4 werewolves and 2 Refuges can dish out an unlimited number of hexes, giving out all the curses, all the coppers, and making your opponents trash a bunch of cards (among other things). I think restricting this down to just action or buy could've made an interesting card though. I think there are plenty of decks that can work with this in interesting ways.
Voyage by Commodore Chuckles
Voyage - Night - $5
End your Night phase. Take another turn after this one where you discard your hand at the end of your Action phase.
A Level 2 extra turn. I think this works well. It gives a restriction to work around to make that extra turn worthwhile. This could work in situations when Tactician would work (virtual coin). It would work very nice with tactician

. Overall I like it.
Penance by setchaflePenance - Event - $4
If the previous turn wasn't yours, take an extra turn after this one. During that turn when you play a card, discard a card sharing a type with the played card or reveal you can't.
A Level 3 extra turn. This sounds a bit harsh to work with. It's almost impossible to grow your handsize (labs or village-smithy don't work), meaning you won't be able to play that many payload cards. Absent of some very key cards, it's going to be hard to produce alot of $ on this turn. In addition, I'm not a fan of not limiting the buys for this.
Frontier by anordinaryman
Frontier - Action - $3
Gain a Copper to the top of your deck per Frontier you have in play. If this is the first time you played a Frontier this turn, and the previous turn wasn't yours, then take an extra turn after this one.
A Level 1 extra turn. Obviously, there's no reason to play more (barring some weird reason to want coppers). It feels unnecessary to even put the "per frontier you have in play" clause. That aside, this sounds a bit too strong. Outpost gives you three cards and costs $5. This gives you 4 cards plus a copper and costs $3. Yes, you get junked, but it shouldn't be too hard to trash it it (consider that trashing is usually necessary in an outpost strategy anyway to guarantee getting reasonable cards in your 3 card draw).
Extra Allowance by majiponiExtra Allowance - Event - $2
Once per turn: Take your Allowance token. If you have 3, remove all tokens used in this game, and if the previous turn wasn't your turn, take another.
This extra turn falls in the other category (you can get it on its own extra turns but it takes 3 for 1). I like the idea of setting aside a little for a turn down the line. I think a +buy would make this a little better (since I can get this early game without wasting my buys). However, I think the "remove all tokens from the game" clause is way too wonky. I mean it doesn't even say your tokens, so this will affect your opponents too. All debt tokens, coffers, villagers, VP tokens (bye-bye lead), journey tokens (how does that even work if you don't have a journey token), vanilla bonuses, etc. get removed which means going for them is a waste in this game. Just removing the allowance tokens would've been fine.
Con Artist by mail-mi
Con Artist - Action - $5
+1 Card
+1 Action
+ $1
At the start of your next turn: +1 Card, +1 Action, +1 Buy
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While this is in play, each other player may take an extra turn at the end of one of their turns if the previous turn wasn't theirs. If they do, at the end of their extra turn, +6 Coffers, +4%, +2 Villagers
A Level 3 extra turn for your opponents at a cost. I feel like the cost of this extra turn is bit steep (this is especially true in a 4p game where you can get the bonus multiple times). I would have to be confident I'm getting more than a province since I'm giving you 4 VP tokens and 6 coffers (which can translate into VP). The card itself also feels pretty strong (peddler this turn, almost a level 3 city next turn). I think this could also be frustrating in a 3p or 4p game where one opponent is utilizing another opponent's con artist when he really shouldn't. I'm not sure if there's a great way to give opponents extra turns in this manner.
Siege by grep
Siege - Project - $5
At the beginning on your turn, if the previous turn was not yours, you can discard your hand to take your Siege token.
At the end of your Cleanup phase, you can return the Siege token. If you do, take another turn after this one.
Level 1 Extra turn. Reminds me of Guide. Often it won't really be giving extra turns, since you'll be using up this turn to get it, but with some shtick (guide, duration bonuses) you could turn this into an extra turn. I like it.
Prototype by NoMoreFunPrototype - Action - $4
Gain a card costing up to $4
If this is the first time you played a Prototype this turn, and the previous turn wasn't yours, you may set aside the gained card to take an extra turn after this one, and put the set aside card into your hand instead of drawing your next hand.
A Level 1 extra turn. I think there's something to the concept of being able to construct your extra turn hand, but I don't think this will usually work as is. There are few $4 or less cards that can start an engine by themselves unless your deck is really well stacked (and I mean really well stacked). Even if there was a card that it could work for, it'd be a little boring, since all players will just use prototype on that card until it's gone and then prototype doesn't work anymore.
Negotiator by Gazbag
Negotiator - Action - $6
If this is the first time you played a Negotiator this turn, and the previous turn wasn't yours, then take an extra turn after this one. During your next turn, whenever you gain a card, each other player gets +1 Villager and +1 Coffers.
A Level 1 extra turn. Nice and simple. An extra turn where your opponents get a bonus for you buying cards (which is usually the point of the extra turn). I like it.
Borrowed Land by 4est
Borrowed Land - Victory - $4
1%
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When you gain or trash this, take your Turn token and return it at the start of Clean Up. If you do, and the previous turn wasn't yours, take another turn after this one during which you cannot buy Victory cards.
A Level 1 extra turn. I think this would've been nicer without the token (even if it is wordy). Regardless, I like the card. You get an extra turn at the cost of junking yourself. In addition the extra turn can't be used for victory cards. Since this also gets use on trashing, I can utilize this even in the beginning since I get a net +buy when I buy and trash this.
Desert Outpost by Gubump
Desert Outpost - Action - $5
+2 Cards
If this is the first time you've played a Desert Outpost this turn and the previous turn wasn't yours, then set aside up to 5 cards from your hand, take an extra turn after this one, and put the set aside cards into your hand instead of drawing your next hand.
A Level 1 extra turn. It feels a bit strong. Even though you have to sacrifice cards this turn, the fact that you can choose them is a big deal. I can set aside 2 cards (village-smithy) and have a reliable engine next turn. And it gives you good odds of having the cards you need with that draw. I think it's a little OP especially at the price point.
Memorial by King Leon
Memorial - Victory - $6
2 VP
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If this is the first time, you bought a Memorial this turn, take an extra turn after this one, during which you can't buy Victory cards.
A Level 2 extra turn for buying a victory card. Not bad. An extra turn without victory cards at the cost of junking yourself a little. I like this kind of balance getting the benefit without junking yourself too much.
Heir by spineflu
Heir - Event - 4d
Once per turn, choose one: Gain a Curse to your Heirs mat, then each other player gets +2%; or put two Curse cards from your Heirs mat onto your deck to take an additional turn after this one.
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Setup: include 3 additional Curses per player in the Supply
This extra turn falls in the other category (you can get it on its own extra turns but it takes 3 for 1). This seems way too expensive for a single extra turn (that only gets 3 cards and 2 curses in hand). It takes $12, (in debt) 3 buys, 4vp to all opponents, and 2 curses to get a single outpost-like turn. You'd have to get a province just to break even on points (not to mention the opportunity cost of $12 and 3 buys). On top of that, you have to set up a deck that can work with outpost. Compare that to just buying Outpost a single time and getting extra turns every round. I also think this will just make cursers more strong in that they will be able to curse longer.
Raft (Traveler Line) by Fly-Eagles-FlyRaft - Action - $2
Trash a card from your hand.
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When you discard this from play, you may exchange it for a Fishing Vessel.
Fishing Vessel - Action - $3*
+ $2
+1 Buy
At the start of your next turn, choose one: + $1, +1 Buy; or you may exchange it for a Cog, putting it on the bottom of your deck.
(This is not in the Supply.)

Cog - Action - $4*
+3 Cards
+1 Action
Discard 2 cards.
At the start of your next turn, choose one: +3 Cards, discard 2 cards; or exchange this for a Galleon, putting it on the bottom of your deck.
(This is not in the Supply)

Galleon - Action - $5*
+1 Card
+1 Action
+ $2
Each other player discards a treasure card from their hand or reveals they can't.
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When you discard this from play, you may exchange it for a Flagship
(This is not in the Supply)

Flagship - Action - $6*
+1 Card
+1 Action
While this is in play, when you play a Duration card, +1%. At the end of the game, choose one:
+2%; or take an extra turn.
(This stays in play; This is not in the Supply)
A Level 0 extra turn. I'll be a little lenient in this judging since Traveler lines are hard to make. Raft is good; you avoided the common pitfall of making the beginning of a traveler line too good. It's weak for $2 (not including the exchange) which is what it should be. Fishing Vessel is also a weakish card (as it should be). Cog is very good if you choose +3 cards for next turn (not sure why the discard option is there). I was thinking this might fit better in slot 4, but then again warrior is usually the card people keep over hero so it's not crazy that the better card is in slot 3. Of course galleon is also very strong. BTW I like the creative way to exchange fishing vessel and cog instead of the simple cleanup way.
My thoughts on Flagship:
1) You get an extra turn per flagship at the end of the game, but how do you know how many you have without looking through your deck? You should probably get a token when you gain a flagship to track this. Also this should be below a line (just like the in play effect). In addition it should be orange.
2) I like the choice between extra turn and 2vp since sometimes 2vp might just be better.
3) The while in play effect can kind of break this. Since you have so many durations in this line alone (15) and you can stack multiple flagships (each which will get points when played), you can really get some broken VP engines with this (especially cog which is a great draw card). Unlike most other VP token cards, this is not contingent on buying/gaining cards or trashing them, and it can give is very likely. It's probably likely even without since if I have 3 flagships and play 3 duration cards per turn, that's 9 points without changing my deck.
Conclusion: There were some pretty good cards to choose from. The top 5 were Double Shift, Voyage, Siege, Negotiator, and Borrrowed Land. This was a pretty tough decision among these. At the end of the day, Negotiator didn't seem like too much of a choice (it seems always worth it to go for). Between the rest I was really drawn to voyage's challenge of making a deck that could ultilize it's extra turn, and Borrowed Land, which required a little balancing with your extra turns. In the end, I'm drawn more towards voyage.
Winner:
Voyage by Commodore ChucklesRunner-Up:
Borrowed Land by 4estCongrats Commodore Chuckles (and everyone else)