My favorite is Kummer's variation of Euclid's proof.
Perhaps the strangest is Fürstenberg's topological
proof. Check them out and see which one you like.
[Move up one level]
Euclid's Proof (c. 300 BC)
Furstenberg's Topological Proof (1955)
Goldbach's Proof (1730)
Kummer's Restatement of Euclid's Proof
Filip Saidak's Proof (2005)