Here's my 4-turn solution
This solution depends on you being player 1 in a multiplayer game.
Turn 1
YOU: $5, Jester
Player 1: $5, Cache
Player 2: $5, Cache
Player 3: $5, Cache
Turn 2
YOU: $2, Duchess
Players 1-3: Nothing
Turn 3
YOU: Jester, gaining 3 copies of Cache
Players 1-3: Irrelevant
Turn 4
Draw Duchess, Cache, Cache, Cache, Estate, buy Colony