I don't listen to /every/ episode of every podcast as they come out, but my phone's new SD card has more than enough space to just grab 'em all anyway, letting me binge whenever I feel like.
My top recommendations - the ones I /do/ listen to every episode immediately - are The Skeptic's Guide to the Universe; Revolutions; Under the Influence; Decoder Ring Theatre; Welcome to Night Vale; and the top five or six of the 'insight porn' list.
They should all be easy enough to find links to; if they're not, just ask and I'll provide 'em.
Law Bytes; The Docket; Lawyered; The McGill Law Journal: All four about Canadian law.
Ham radio podcasts:
ARRL The Doctor is In
ARRL Audio News
This Week in Amateur Radio
Sciencey shows:
Science Friday
This Week in Science
Astronomy Cast
Quirks and Quarks
Everything Hertz
Rationalism shows:
The Skeptics' Guide to the Universe: If there's one podcast that started pointing me in the right direction to clear my thought processes, it's this one. Getting a feel for this level of skepticism helped prepare me to get a good foundation for figuring out more difficult truths, and trickier ways of telling truths from falsehoods.
Harry Potter and the Methods of Rationality: Goes through the fanfiction HPMOR, and then a few other stories of the same general theme.
Tsuyoku Naritai!
The Bayesian Conspiracy
Rattle Fiction Podcast
Rationally Writing
History shows:
Revolutions: Listen to this one from the beginning. In an engaging and non-boring way, the host covers ten of the most important revolutions in history. A lot of the modern world makes a lot more sense after hearing about the sharp-left-turns events took to get here.
Dan Carlin's Hardcore History
You Must Remember This
The Podcast Very Different From Ours: A reading of the book "Law Systems Very Different From Ours", a good resource for anyone who wants to worldbuild a setting that's interesting, self-consistent, and noticeably distinct from the present-day.
Cautionary Tales
Music podcasts:
Irish and Celtic Music Podcast
Monthly Best of Electro Swing
"Insight Porn": Each episode is usually a different topic, and after you listen to each one, you have the feeling that you've learned something, even if you forget it all by the time next week's episode rolls around.
Radiolab
Freakonomics Radio
Invisibilia
99% Invisible
Futility Closet
Reply All
More or Less: Behind the Statistics
TED Talks Daily
Stuff You Should Know; Stuff To Blow Your Mind; Stuff You Missed in History Class; Stuff Mom Never Told You; Stuff They Don't Want You To Know
Fiction:
Decoder Ring Theatre: The "Red Panda Adventures" is one of the best pulp-era comic-book universes, bar none. Their other productions aren't too shabby, either.
Welcome to Night Vale: Hard to give a good description of, but worth listening to from the first episode.
Hello From The Magic Tavern: Improv fantasy, of the "Yes, and..." worldbuilding style, and a lack of shame that frequently hits NSFW levels.
PodCastle; PseudoPod; Escape Pod: Regular short stories of fantasy, horror, and science-fiction, respectively.
Mission to Zyxx: Feels a lot like 'Magic Tavern', except sci-fi.
Improvised Star Trek
Clarkesworld Magazine
Random individual topics:
FurCast: The only still-running furry podcast I could find.
Under the Influence: All about the marketing industry, by an ad professional. Much better than that sounds, and a lot of handy lessons about advertising-manipulation tricks to watch out for. Worth hitting the archives for; used to be called 'Age of Persuasion'.
Savage Lovecast: Relationship and sex advice from Dan Savage.
Democracy Now! Audio: Daily politics and news.
Ear Hustle: Describes what life is actually like behind bars.
Off the Hook; Off the Wall: Produced by 2600, the magazine for white-hat hackers.
And for readers
Furcast - http://furcast.fm/
Link
D'Otter
Sorry, I don't hang around this site very often.
The Anhroholics - https://soundcloud.com/user-534967572
Fangs & Fonts - http://www.fangsandfonts.com/
...hasn't updated in a while, but back episodes might be interesting to a writer
Bearly Furcasting - https://overcast.fm/itunes1511338426/bearly-furcasting-feat-taebyn
...new
Fur What It's Worth - http://www.furwhatitsworth.com/
Southpaws - https://southpawscast.podbean.com/
...was Knotcast
Friday Night Tracks - https://fridaynighttracks.com/
...occasional (~monthly) EDM podcast by Paradox from Furcast