Newcomb's Cat: There's an opaque box containing a cat and some poison gas. There's also a transparent box, which you can see contains a live kitten. Depending on how you're about to decide, the poison gas might or might not have been released in the opaque box, and as a result you're in a superposition of states, mysteriously all of which involve cleaning up one sort of cat-related mess or another.

Pascal's Dilemma: There are two heretics imprisoned by the Inquisition, and they separately have to decide whether or not to recant and believe in God. The terms of the problem are set up so that it's invariably in each prisoner's selfish interest to believe in God, and yet for some reason they both turn out to decide it's all a load of rubbish.

The Chinese Demon: There's a room in which people are shuffling pieces of paper in the execution of a complex algorithm that answers questions in Chinese. However, the door to the room is guarded by an infinitesimal demon that only lets the most difficult questions through, and as a result the contents of the room become more and more disordered until nobody can find the pieces of paper they want any more.

(I accidentally invented one of these in the pub just now, and thought it was fun enough that I should add a couple more and post it.)