App is downloaded.

Something educative. Along the lines of R4… history, geography, culture, etc….

This American Life
Lots and lots of good true crime stuff – True Crime Garage is good.
Museum of Curiosity
Infinite Monkey Cage
The fifty nine things that made the modern economy thing.

Stuff You Should Know

Myths and Legends https://www.mythpodcast.com/
Freakonomics http://freakonomics.com/archive/
The Moth Radio Hour https://themoth.org/radio-hour
99% Invisible http://99percentinvisible.org/episodes/

Dan Carlin does a great WW1 podcast
http://www.dancarlin.com/product-tag/world-war-one/

Untold: Daniel Morgan Murder.
Desert Island Discs!

Lore, 99% Invisible.

And My Dad Wrote A Porno, obviously.

In Our Time (Melvin Bragg)

Answer Me This

No such thing as a fish…The QI researchers Podcast very funny and informative.

Without replicating too many of the above suggestions:

Reply All – Similar to This American Life
S-Town – A short true life series that comes across like a novel, brilliantly put together.
Revisionist History – Malcolm Gladwell’s project, bit like the best of Freakonomics.
The Butterfly Effect – Jon Ronson look at tech and porn. Not available ’till the Autumn but looks like it’ll be interesting
Longform – Specifically about journalism but can be interesting if you pick and choose a bit.
The Guilty Feminist – Mostly female perspective on social issues and feminism
Ear Hustle – Life inside San Quentin prison, produced by prisoners.
The Reith Lectures – Recordings going back to the 1940’s

More general chat show format podcasts, how informative they are depends a lot on the guests
Marc Maron’s WTF podcast
An Irishman Abroad
Heres The Thing with Alec Baldwin
The Comedians Comedian – Discusses the process of comedy, normally more interesting than funny.
The Adam Buxton Podcast

More light hearted ones.
Richard herring Leicester Square Theatre Podcast – Interviews but generally more comic than informative
Beef and Dairy Network – Surreal comedy based on the beef industry (but nothing to do with it really)
The Bugle – Andy Zaltzman topical news podcast

If I had to recommend just one podcast it would be S-Town.

“Say Why To Drugs” and “Stop And Search” might be of interest. Also “Distraction Pieces” is usually good, interviews and more entertainment than informative but there are loads to pick from.

From our own correspondent.
Infinite monkey cage
In our time.