My Current List:
Long Form
Adam Ruins Everything - Long form interviews with the experts featured on the show
Hardcore History - LONG podcasts that go in depth into various periods and people in history. A recent "blitz edition" podcast was 6 hours. Be prepared!
Dan Carlin's Common Sense - Political podcast by the Hardcore History host, currently in a bit of hiatus due to, well, you know, while he processes.
Ezra Klein - Long interviews with whoever Ezra Klein wants to interview.
Politics
The Weeds - Podcast from Vox, goes in-depth into the data surrounding the issues.
AEI Banter - From conservative thinktank AEI, if you're interested in classical, non-reactionary conservative thought, it's pretty decent.
538 - From the Website. Focuses on statistics and polls.
Slate's Political Gabfest - General political discussion. Tends to be very even-handed, even if it does lean left.
What Trump Can Teach Us About Con Law - 99% Invisible guy interviews a constitutional law professor about what's happening in the headlines.
Slow Burn: A Podcast About Watergate - Things just don't change, do they
TV
West Wing Weekly - Rewatch of the show, featuring interviews with cast members and experts in the issues that the show discussed
Watching Dead - A podcast by two guys watching the Walking Dead and are Done. I don't watch anymore, but still like to hear them rip the show to shreds.
Game of Thrones Podcast - Breaks down the episodes.
Religion
Phil Vischer - Three Evangelicals discuss current events, why Christian art sucks, why pop culture isn't bad, and the current state of Evangelicalism.
Relevant - If Christianity Today is the magazine that all the Baby Boom/Graham era Evangelicals read, Relevant is the magazine that all their children are reading. If you want to know where Evangelicalism is going and what the next generation of leaders think, and if you suspect you might not be getting that from the news (you're not), this is the podcast to listen to.
Culture
Reply All - Podcast on the Internet and the culture surrounding the internet. Yes-Yes-No is always hilarious.
This American Life - Classic. Does it need an explanation?
Crimetown - In depth storytelling about the corruption and crime in Providence, RI
Science Vs. - Australian host breaks down scientific studies on hot issues, so that you don't have to.
Misc
Judge John Hodgman - Comedy podcast where John Hodgman handles disputes between multiple parties. Everyone should know both established law and what is and is not a sandwich.
99% Invisible - Short form podcast about design and the stories behind design. Absolutely fascinating.
Intelligence Squared - Debates on a variety of topics.
Heavyweight - Jonathan Goldstein (from This American Life) tries to help people work through their issues and their relationships with varying success. Some of the most moving things I've ever listened to.