There is a machine with two buttons. Inside the machine is one million one dollar bills. You and another person take turns pushing buttons. If button A is pushed, the machine spits out $1 (if it still has any left) for the pusher. If button B is pressed, the machine spits out $2 (if it still has that much left), and promptly breaks, allowing no more button-pushing for anyone. All the money inside, of course goes up in smoke, because anyone who has watched any science fiction at all knows that machines emit smoke and sparks and fire whenever anything goes wrong. "She canna spit out enna more than a dollar atta time, cap'n!" or something like that.
The weird thing is that a "rational agent", who seeks to maximize their own gain, and assumes that the other participant is rational also, will push the $2-and-explode button on their very first turn, committing $999,998 to the flames. This is true by a simple induction argument: suppose the machine has just $2 left and it is your turn. Of course you will then push the $2 button. If it has $3, then you reason that if you only take $1, your adversary will take the other $2, so you might as well take $2 now. In general, if the machine has $n, and the inductive hypothesis that a rational agent will push the $2 button on any amount less than $n remaining in the machine, then the "rational" choice is to hit the $2 button, because you know your adversary, being a rational agent, will break the machine on their turn if they get one, so you might as well get $2 instead of $1 on your last turn.
I really like this example because it doesn't require the simultaneity of the prisoner's dilemma to produce what seems like extremely bad behavior which is still rational in the economic sense.
GENERAL DISCLAIMER: I don't mean to say that I think that "rational" in the above sense is actually "good reasoning" necessarily, which is, I think, exactly what the example illustrates. Then again, IANAgame-theorist.