Sometimes called Banach-Tarski Paradox.
Theorem: The closed unit ball (the interior of the sphere of radius 1 and surface) in R3 can be decomposed into a finite number of sets, which can be reassembled to 2 unit balls.
This is not nonsense: due to the axiom of choice you have sets, which have no measureable volume. So decompose into such sets and you can reassemble them to sets with a larger volume.
This won't work if you don't believe in the axiom of choice, but I've met no-one yet who doesn't.
I think the number of pieces is 5, works with 6 pieces, too.