The Ross-Littlewood Paradox is amusing.

You have an infinite collection of balls in a storeroom, labeled with the natural numbers (1, 2, 3, etc.) and a vase that can hold any number, or all, of them.

At each integer time T, starting with T=1, you take the 10 lowest numbered balls out of the storeroom and put them in the vase, and then take the lowest numbered ball out of the vase and destroy it. So at any finite time T, there are 9T balls in the vase and all the balls labeled with a number less than or equal to T have been destroyed.

Now, because the number of balls at any given time T is given by 9T, in the limit as T approaches infinity, there are infinitely many balls in the vase. On the other hand, because every ball has a time T at which it will be destroyed, the limit of the set of balls in the vase as T approaches infinity is the empty set. So at T = infinity, you have an empty vase that contains infinitely many balls.

The moral of the story is to be careful what limits you take, because taking two different limits can give two different answers even if they seem like they're measuring the same thing.

(Can this be used as an argument for the existence of nonstandard numbers?)