Theseus' Paradox
Also known as the Ship of Theseus is a famous paradox and one of my personal favourites. Here's the actual paradox straigth from wikipedia.
The ship wherein Theseus and the youth of Athens returned [from Crete] had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place, insomuch that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.
It may seem just like a little brain-scratcher at first but this paradox is almost diabolically important and significant. Humans are not exempt from being in the "things that grow" category. Our entire body regenerates every ten to twelve years (yes, even your brain cells.) So, are you same the same person?