Though technically a 1950s show (the pilot episode was September 29, 1959), I'm including it here because A. There are no other shows in this category yet, and B. it was a 1960's show, so there.
The main thing I remember about this show is that it was so surreal. Way ahead of its time, in some regards. The fact that Dobie's bizarre dream sequences were peopled by future stars such as Tuesday Weld and Warren Beaty just added to the strangeness, in my humble opinion.
Dwayne Hickman was such a likeable guy-next-door, and everybody loved his lazy beatnik pal Maynard G. Krebbs. (I always thought of Bob Denver as Maynard, even during the endless reruns of "Gilligan's Island" - I just thought of it as Maynard playing Gilligan.)
Those rumors about Zelda turned out to be true, by the way. (Yep, she's a Democrat.)
Snag the DVD before they're all gone. (There was only one left, at press time.)