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[humcalc216]

First, the less mathematical/less precise/TLDR version: How many cards can you gain in a turn, starting with an n-card deck all in your hand?

Mathematical Definition/Precise Rules:
Define a function f(n) as follows:

• Set up a legal two-player Dominion kingdom.
• Don't give out any starting cards (so there are 60 coppers in the Supply).
• Select n cards from the supply and/or available non-supply cards for that kingdom, and put those n cards into your hand (and you have no other cards in your deck).
• Your opponent has no cards whatsoever, and the trash begins empty* (EDIT).
• Play a turn.  f(n) is the maximum number of cards you can gain in that turn.
• To avoid infinite loops, a "gain" only counts the first time you gain a particular, physical card in the turn, and it doesn't count at all if you started with that physical card in your hand. (EDIT)

*aside from possible Zombies

Extra restrictions/clarifications:

• The turn you take is effectively the first turn of the game.  So, you have no Durations in play, cards on mats, tokens on piles, etc.  And your opponent did nothing on their last turn (matters for Smugglers, Treasure Hunter).
• To avoid extra shenanigans, if Black Market is in the Kingdom, the Black Market deck is automatically one copy of each card removed from a First Edition.  These cards cannot be used otherwise.  (In my opinion, this should become a standard restriction when Black Market's mechanic is wanted, but its shenanigans are not desired.)
• Shelters can only be used if there's at least one Dark Ages card in the Kingdom.  If Shelters are used, you may only have up to one of each Shelter among the n cards, unless the Kingdom would allow something else to happen (e.g. Masquerade or Lurker+Trasher).
• Heirlooms are treated similarly to Shelters (as are Ghost/Wish if their respective Heirlooms are the only ways to gain them).
• Platinum/Colony can only be present if at least one Prosperity card is in the Kingdom.
• At most two Events/Landmarks.

The questions:

• What are f(n) for some small values?
• For which n is f(n) maximized, and what is the maximum?  (Eventually, f(n)=0, so this has to exist.  Presumably, it will be an n for which you can empty the supply and possibly non-supply, and the board has sufficiently many cards.)

Some examples:

• n=0.  Alms for Death Cart gains 3 cards.  So, f(0) is at least 3.
• n=1.  Start with an Engineer.  Engineer for Death Cart, trash it for Death Cart, Alms for Death Cart gains 9 cards.  So, f(1) is at least 9.

[Awaclus]

The maximum is unbounded because you can keep gaining the same cards from the trash and then trashing them over and over again. I'm pretty sure we can do that with n=2 using this and Diadem (Tournament can replace Pooka or Tracker, as those are only used for their Heirlooms which we don't need because of Diadem).

[humcalc216]

The maximum is unbounded because you can keep gaining the same cards from the trash and then trashing them over and over again. I'm pretty sure we can do that with n=2 using this and Diadem (Tournament can replace Pooka or Tracker, as those are only used for their Heirlooms which we don't need because of Diadem).

You're right; of course I forgot something :/.  I had meant to only count a gain the first time a physical card is gained in a turn, and I've edited the original post to reflect this.

But, without that change, I guess f(2) is at least 276 plus however many Wisps and non-Diadem Prizes you can gain on this board (probably all of them).  Though, f(n) has to be greater than or equal to f(n-1) for some n>2, since you can have Kingdoms with more available cards than this one.

[faust]

For n=0, there are various ways to get to 4:
- Alms for Death Cart, exchange for Changeling.
- Alms for Villa, play Villa, buy Borrow, use Baker token, buy Stonemason overpaying by $1 to gain 2 Poor Houses.

If you had space for at least 3 Events, you could get insane with Villa+Borrow/Travelling Fair/Advance.

Random question on the "trash begins empty" edit: Does that mean we cannot use Necromancer or would Necromancer just not spawn any Zombies?

[humcalc216]

Random question on the "trash begins empty" edit: Does that mean we cannot use Necromancer or would Necromancer just not spawn any Zombies?

Yes. Zombies begin in the Trash if Necromancer is in the Supply.  (Too many edge cases to keep track of!)

[majiponi]

Just have many Coin tokens, trash your deck with Raze. Travelling Fair enables you to gain everything.
I think banning startup is needed.

[faust]

Just have many Coin tokens, trash your deck with Raze. Travelling Fair enables you to gain everything.
I think banning startup is needed.

It is:

• The turn you take is effectively the first turn of the game.  So, you have no Durations in play, cards on mats, tokens on piles, etc.  And your opponent did nothing on their last turn (matters for Smugglers, Treasure Hunter).

[majiponi]

Just have many Coin tokens, trash your deck with Raze. Travelling Fair enables you to gain everything.
I think banning startup is needed.

It is:

• The turn you take is effectively the first turn of the game.  So, you have no Durations in play, cards on mats, tokens on piles, etc.  And your opponent did nothing on their last turn (matters for Smugglers, Treasure Hunter).

Oh, I didn't notice. Then, I'll say f(2) is very huge. At least 30, I guess. (Hint: my puzzle thread)