The Mountain vs. The Hound
Season 1, Episode 5
Why they fought: The Mountain flipped out after losing a tourney battle to Ser Loras. The Hound jumped in and saved Loras' life (but wasn't fast enough to save that poor decapitated horse).
Winner: King Robert stopped the fight before either Clegane could claim victory, but The Hound impressively held his own against his brother, arguably the most feared fighter in Westeros.