What combination of kingdom cards makes for the fewest number of turns to end the game?
You can end the game, even buying/gaining every single card in the entire supply, on turn 1! There are multiple possible ways to do this, but here is one:

The Kingdom:


Fortress
Villa
Pixie
Pooka
Tracker
Overlord
Lurker
King's Court
Bridge
Library
Travelling Fair
Advance


I'm not going to go into all of the details here, because it gets quite long and complex, but the final step is to have bought a Travelling Fair, and a Villa to return to action phase. Then have a hand of:

King's Court, King's Court, Lurker, Lurker, Overlord


Play King's Court (a)


Play King's Court (1)


Play Lurker (1), trash Overlord from supply
Play Lurker (2), gain Overlord from the trash (put it on your deck because of Travelling Fair)
Play Lurker (3), do anything, doesn't matter

Play King's Court (2)


Play Overlord(1) as Pixie, draw next Overlord, trash it to get the Boon
Play Overlord(2) as Lurker, gain Overlord from the trash
Play Overlord(3) as King's Court (a)




That last step wraps back to the first... you have the same hand, same card on top of your deck, same cards in the trash. Except you have received a Boon. 

With the infinite Boons, you will have received every Silver in the game from The Mountain's Gift, drawn them all into hand from The Sea's Gift, and gained all of the cards with The Earth's Gift.

The full details of this combo are written out in this forum post. All credit goes to Mith and other f.ds members. 

There is also a video of this combo being demonstrated on Dominion Online:

https://www.youtube.com/watch?v=pbWdV9H2SKk



So that particular combination requires a very convoluted setup and an extremely lucky starting hand, with the exact right Heirlooms. So here's a very short game that happened in a real, regular randomized game:

Turn 3, draw Quarry + 4 Copper.
Stonemason -> 2 Villas (5 times).
Stonemason -> 2 Trade Route (4-5 times).
Win off Museum points

Empties the Villa, Stonemason, and Trade Route piles to end the game.

Details in this forum post.