As black, it's possible to win a game of Chess in two moves. This play even has a name, Fool's mate, as it relies on the white player's poor judgement. Now Monopoly doesn't lend itself to such a predictable losing strategy, given its randomness, but with the correct alignment of celestial bodies, it's entirely possible to bankrupt a player in a mere four turns.
Scatterplot's Dan Myers decided to orchestrate the feat on video, which you can watch above. It doesn't do that great a job of demonstrating how the hypothetical game would play out, but, thankfully, Myers has outlined the steps on his blog (using the game's US edition). Here's a snippet of the first two turns, so you can get an idea of its improbability:
Player 1, Turn 1:
Roll: 6-6, Lands on: Electric Company Action: None, Doubles therefore roll again
Roll: 6-6, Lands on: Illinois Avenue Action: None, Doubles therefore roll again
Roll: 4-5, Lands on: Community Chest "Bank error in your favor, Collect $200" Action: Collects $200 (now has $1700)
Player 2, Turn 1:
Roll: 2-2, Lands on: Income Tax Action: Pay $200 (now has $1300), Doubles therefore rolls again
Roll: 5-6, Lands on: Pennsylvania Rail Road Action: None
The odds of this magical sequence occurring is beyond remote, considering it requires specific dice rolls, consecutive moves/actions by the one player, a set ordering of Chance and Community cards and of course, player cooperation.
It's also been pointed out that by the rules, if a player decides not to buy a property he or she lands on, it goes to auction, where it could be sold for a single dollar. While both players can choose not to bid, it's highly unlikely.
Fun exercise, though. Any suggestions for other board games?