I've got quite a variety:
loads of Radio 4 programmes
1A
99% invisible - interesting programmes in themselves, but I could listen to Roman Mars reading the telephone directory. I love his voice.
Ear Hustle
Embedded - going behind news stories
Freakonomics
Hidden Brain
Invisibilia
Kermode and Mayo's Film Review
Rhod Gilbert's Best Bits - chatting about nothing, good for when I want something light to listen to
Simon Mayo's Confessions - the only podcast that regularly has me laughing out loud
Sincerely,X - from TED radio, but stories that need to be told anonymously
Stuff to Blow Your Mind
Stuff you missed in history class
Stuff you should know
TED radio hour
The Bugle - topical comedy
The Guardian's Audio Long Reads
The Moth - people telling stories on stage in story slams. Huge variety and range from the hilarious to the extremely moving. Absolutely brilliant
The Allusionist - about words and language