Godzilla's toughest opponent? Destoroyah, no doubt. He is the Oxygen Destroyer with form, after all. True the military can take him down with cold weapons, but this isn't about who is OUR toughest opponent (though that would probably be... let's see... Godzilla! XD), it's about Godzilla's tough guys. Either him or Bagan, but I'm lead to assume this is strictly film speaking here. And I don't go by fan fiction, either.
Showa King Ghidorah is another good one, too. As someone has already noted, Godzilla never took down Ghidorah without a partner at his side. So it was always Godzilla and another monster vs. Ghidorah (whether he has allies or not doesn't matter). Hedorah is also an intriguing one, seeing on how it took human assistance to take him down, like Destoroyah. MG and MG2 are pretty brutal in their own right. Can't really say Kiryu, but he has the Absolute Zero Cannon.