Here's an insanely skilled Diablo III player Dat Modz proving that it's possible to get to max level in no time. Sixty-five seconds to be exact. That's bloody impressive.
And how is this possible? Dat Modz is on the game's very rare (not THAT rare since patch 2.2, but still rare) Cow Level, and its special event gives a massive XP boost. Also his character is wearing a Hellfire Ring with the Gem of Ease and equips a Leoric's Signet as soon as he hits the appropriate level. All of these give another huge XP boost. That Ruby in the helmet's socket and the extra Paragon Levels help too.
Here's the video showing his Monk, the SSJ3 Goku theme from Dragon Ball fits perfectly:
The pop-ups can't even keep up with gaining levels this fast, but you can track the increases in the top left, above his avatar.