I'd bet 11 turns is possible. I bet I could do 12. Just use my solution to Lucky Chancellor, but switch out Moat, Chancellor, and Fortress for Watchtower, Squire, and Scrying Pool. Eh... maybee you need Haggler, so put BM (with Baker and YW) and make one of them the bane. Then buy a bunch of actions turn 10-11, but with a Watchtower in hand to trash a squire and topdeck a SP.
Hmm... It might actually work with so few turns (but you're assuming having a 5/2-split, right?), but I still see some problems.
If I can count, at the start of the game the supply contains 303 cards (assuming 1-player game, colonies and YW in supply) (Treasures: 53C+40S+30G+12Plat+16Pot, Victory Cards: 8*4, 10 Curses, 110 Kingdom cards). We don't want to play to much bridges, otherwise our hagglers don't gain cards, but we want to play some, to make things payable. I think 7 bridges is the ideal amount, then you can gain a card with haggler for every Province, Colony and Platinum buy. If we end our mega-turn with 10 hagglers in play, have 7 cost reduction, we can probably manage with 38 buys and 52 coins (maybe we need a few buys and coins more).
That way you can buy all victory cards and 6 platinum, gaining all other cards with haggler (except Scrying Pool, which should be gained by trashing all gained Squires immediately with a Watchtower). Then you buy 38 cards, gain 220 cards with haggler, and ~9 cards with Watchtower+Squire, totalling 267 gained cards in our megaturn. So in this scenario we will need to gain the other 36 cards in earlier turns.
Coins won't be the problem, but getting enough buys will be a problem. We can't use Squire for +buys, because they all need to be trashed to gain the Scrying Pools, and we just get 7 buys from bridges. So I think we should add a cheap source of +buy in the kingdom, like Candlestick Maker. If you king court all 10 of them, and play 7 bridges, you get 38 buys.
So if we follow this idea we will need to start our mega turn with the following cards (or gain them during our megaturn):
-7 King Court (giving 13 positions, 10 for CM, 2 for bridge, also giving enough actions to play hagglers)
-3 Bridge
-10 Candlestick Makers
-10 Hagglers
-Scrying Pool (in starting hand)
- Watchtower
- Some other action cards (no non-action cards must be in deck)
Note that this deck also fulfills the requirement that we gain 36 cards on turns before our mega-turn (we have also gained Ch, Fg*2, Mint, Squire, totalling 3)
Probably we want to make a mini-megaturn before our real megaturn to get all needed components. For example with the following deck we can gain all components:
- 2 KC
- 2 CM
- 2 Bridge
- 2 Hag
- SP (in starting hand)
- WT
Then you can play SP -- KC-KC-Br-Br-CM -- Hag -- Hag, buy 8 KC, Bridge and Squire (trash for SP and topdeck), and gain 8 Hagglers and 8 CM.
This seems to point in the good direction for a solution...
By the way: I'm not expecting any solutions of this puzzle, because with worst possible shuffle luck this puzzle is very hard to do efficiently, and probably the optimal solution is very tedious to write down. But I could be wrong.
Well, it depends on what qualifies as a solution. I'm guessing that "good" solutions might not ever come up, but it's not too difficult to find just one solution. For example, if you can just find a way to trash your deck down to Salvager and Province (and up to three other cards), you can buy one card at a time and finish in probably less than 300 turns. If you can set up Village-Salvager-Salvager-Province-Province, that's probably less than 200 turns. Other people might be able to come up with better solutions. Whether a "good" solution exists is probably dependent on whether it's possible to make a supply-draining megaturn starting from just 5 cards, and I wouldn't be too surprised if it is.
Certainly with worst-case shuffle luck puzzles, you want to trash down as quickly as you can so that you don't have to deal with lots of cases. I'm guessing that finding a good solution (15 turns or less or so) will require two main steps: 1. proving that some trashing method can reach the desired deck in a given number of turns, and 2. setting up a megaturn starting from 5 cards.
I was talking about good solutions indeed. Note that your suggestions do not quite work, because you will three pile before you empty the supply. In the last turn at the very least 15 cards must be gained.