my latest is kayle, i'm probably better with her than i am most characters so i know how to play her decently well.
my build was as follows: madreds bloodrazer, chyrstal sceptar, sorcery boots, nashors tooth, frozen heart, and guinsoo's rageblade.
summoners: heal, ignite
ap, att. sp, armor runes
with heal, ignite and kayles ult + heal at max CDR i didnt need buffs but with was a close call and the max CDR was a must
hextech gunblade may be a better option than guinsoo's rageblade but idk thats just how i built her

so if you have a good story of how you killed baron by yourself using some off the wall champion feel free to share it

Is Irelia unlikely? Because I've done it with her ;P

I've soloed it as ap fizz before...That was tricky.

Eve.

I've done it as Leblanc. Lots of spell vamp.

Irelia...

Yep.

I saw a jax do it, a master yi and a kog'maw, but.... Ehh. I've never bothered with it too much. Seems pointless.

Soraka.. before they made it impossible to reduce mr. With blue buff/spellvamp and enough ap you can pull it off.

Probably not unlikely but it is easy to solo it with Morde with AP items and spell vamp. Master of killing minions and monsters.

solo him pretty regularly with riven but i doubt people would find that unlikely