Part of my love for Third admittedly comes from information outside of the show itself, namely Jon Pertwee working with Bond creator Ian Fleming in the Naval Intelligence Division during WWII. Reading him describe what he did there, it sounds an awful lot like he was (at least in part) an inspiration for Bond himself with his love for gadgets. Between that and the Sherlock Holmes cribbing during his era, with the Master as his Moriarty, I see his era as representing a culmination of the best of British fiction, between Bond, Holmes, and Doctor Who itself.
As a character, I wish I could attempt insight into what I like about him; I just think he’s cool. The alien defender of Earth, facing off against environmental threats, and while he could whip out some Venusian aikido in a pinch, I appreciated him more as the Diplomatic Doctor. The one who would push himself and those around him to find the peaceful solution to a conflict, and be extremely saddened if it didn’t turn out that way.
His relationship with UNIT is fascinating to me too. He definitely disagrees with some of their methods, and he has some particularly Holmesian moments where he’s exasperated at being the smartest (and often most ignored) man in the room, but it’s a matter of necessity without the TARDIS. As much as this era, for the most part, went against the main drive of the show (being able to go anywhere and anywhen), it made up for that with how much character work it put in place for the Third Doctor and his companions. And yes, I’m counting the Brigadier as a companion in this case; he and Third had terrific rapport with each other.
He definitely has one of my favourite regeneration stories too in Planet Of The Spiders. Not only does it play into the hubris of his particular incarnation, it was something that was actively set up and built upon throughout his era. The clarity with which his want for knowledge would lead to his end, and realising that he really does consider Earth to be his second home, is part of the reason why I still struggle with the notion that Tenth has a tragic story to his character, as it wishes it came through as well as that did. Also, crazy showcases of vehicle chases, gadgets, and regeneration examined through Buddhism, which I think is neat.
While Eleventh will always be my Doctor, Third comes a close second. I could watch Pertwee as the Doctor in just about anything and get into it, even Dimensions In Time (okay, only the parts with him in it, granted, but point still stands).