For my favorite immortal or outsider monster, I think it's got to be some variety of devil, probably the pit fiend. It's got the classic devil form – horns, wings, red skin, massive musculature – and it's intelligent and utterly malevolent. They're smart enough to be the evil mastermind, but also strong enough to go toe-to-toe in melee combat.
The fun part about pit fiends and other devils is not using them as just another foe. Pit fiends are the masters of the other devils. They should be major antagonists over the course of entire campaigns. At low levels, the PCs might face a devil cult with pit fiend imagery in the cultists' lair. At medium levels, the pit fiend may send lesser devils after the group, either to destroy or corrupt them. At higher levels, the PCs can finally track down the pit fiend on its own plane to try to find a way to permanently kill it.
The pit fiend is not only the classic devil, but if it is played well, it can be an enemy that the players remember for years afterward.