One of our fellow summoners, Descyphal, has made up a great playlist on YouTube showing currently up to 57 videos of champions that have been able to solo Baron Nashor. Go check it out, give suggestions for what champions to try next, and subscribe!
https://www.youtube.com/watch?v=VGlNU3EomyQ&index=2&list=PLIMNqEVHRrMaHvUOsGoLiz4pUIIGWbo0L
LOL nice!
My favorites are Mundo, Teemo, Singed, and Sion. Of them, only Sion can solo baron, I am quite sure XD
Let's see 
now.
All champions can solo Baron, it just comes down to a matter of out-sustaining the damage he dishes out - which is only healing something like ~100 HP per second, not super hard to accomplish with life steal alone.
With their standard build you're going to have issues unless you're playing a jungler or duelists designed to win extended brawls, but everyone can build like them.
Anyways, I've done it plenty of times on Ashe (before her %AS per level was buffed). Her base AD is actually notably lower than average, causing a lot of mages and supports to bypass her in base stats (and thus ultimately scale better with pure damage vs. Baron). Her auto-attack steroid also basically doesn't apply to Baron Nashor either, so if It's possible with a no steroid champion with below average base stats, it's not a stretch to assume every champion with an ADC path can do it. I'd also like to point out that having no AA steroids is actually pretty rare - "mage" is the only archetype that doesn't come with an AA steroid more than 50% of the time. Tanks, Fighters, Assassins, Marksman, and Supports tend to come with them. Supports tend to have really good steroids to - Janna's +50 AD steroid isn't exactly tickling things to death.
Technically any champion can solo Baron if you build AS-lifesteal. (Tested with AD anivia)
