I know Fiddle can solo it low level as well due to his drain and sustain. I always make sure I have dragon warded when these champions are jungle on the other team
Fiddle can infact solo baron. I've doneit and Throatslasher did it in a game I was in 2 days ago. You need seekers arm gaurd and Spirit of the Spectral Wraith at the very least. Baron puts down bubbles where the knock up is coming. All you have to do is move when you see the green bubbles under your feet.
nunu can solo drake at level 3, so he is by far the best one imo. other than that, i would say generally champions with great clear time like Dr. Mundo(the disadvantage of him is the negative sustain though)
When I'd play Tryndamere jungle, I was able to clear dragon as soon as I'd buy Wriggle's Lantern. (Around the 9 minute mark) I have also solo'd dragon with Lee Sin with just a Wriggle's.
and he can solo baron with lvl 12-13 only needed is 25-30% cdr and a blue buff for more drain
---> Making your clone attack first then attack dragon yourself from behind to proc your passive.