Blow your mind at this page of wikipedia dedicated to all the paradoxes ever dreamed up:
My top six favourites include:
Paradox of the Court: A law student agrees to pay his teacher after winning his first case. The teacher then sues the student (who has not yet won a case) for payment.
Ship of Theseus It seems like you can replace any component of a ship, and it will still be the same ship. So you can replace them all, one at a time, and it will still be the same ship. But then you can take all the original pieces, and assemble them into a ship. That, too, is the same ship with which you started.
Hardy’s paradox: How can we make inferences about past events that we haven’t observed while at the same time acknowledge that the act of observing it affects the reality we are inferring to?
Paradox of enrichment: Increasing the food available to an ecosystem may lead to instability, and even to extinction.
Icarus paradox: Some businesses bring about their own downfall through their own successes.
Hegel We learn from history that we do not learn from history.
Ecology of Knowledge 
These are really brilliant – I hadn’t noticed this page before. .
My favourite is Russell’s paradox; particularly the predicate version. P(x) =defn ~x(x)
The barber one mentioned here isn’t a paradox because it doesn’t lead to a contradiction. No such barber can exist.