So I posted this innocent idea of how the humans could win by, maybe, the 6th player-turn of BSG..
I won't re-quote it here, you can go read it.
The response for the fastest Cylon win on the first player's first turn was equally sane; perhaps even more sane and less convoluted; it could probably happen once in a thousand games.
But pretty insanely, a player figured out how the humans could (hypothetically, purely mechanics-wise) get a Win on the first player's first turn. I'm in awe. It involves the fact that State of Emergency doesn't limit you from playing more State of Emergency on your turn.
So, if you manage to shrink the Leadership deck down small enough (with, say, overloading a check and having Adama draw and keep all the green cards), and trigger enough card-draws from it, you can effectively keep drawing and re-playing State of Emergency, getting an effective infinite number of player-actions, including a Cylon on Caprica drawing crises (it requires losing a food, but you can lose a food at 0 and keep going; you just need to gain 1 food back before the loop ends).
That's pretty awesome, even though it will never happen in practice.