Several of my favorite podcasts made the list:
This American Life
Serial
Reply All
Radiolab
99% Invisible
Freakonomics
Hidden Brain
Conan O’Brien Needs a Friend
WTF with Marc Maron
Song Exploder

Some other podcasts that I recommend (where you can just jump in and not previously have listened):

Cautionary Tales
Omnibus!
Radio Atlantic
The Accused

You can view my complete podcast subscription list at Podstand.