I'll start... All of the links below are to the shows' websites, but you can find them on any major podcast app (e.g. iTunes, Stitcher, Google, Spotify, etc.).
NPR Planet Money - breaks down complicated economic subjects in understandable parts and makes it fun. They do a great job of showing how markets and economics impact our lives, past, and futures. Their topics are very eclectic cover big world-wide issues to local small towns. Two years ago I would never have imagined I would enjoy a show about economics so much, but it's fantastic. There's not really a bad episode; just scan through the titles/descriptions for a subject you are mildly interested in and listen... you will come back for more. If you want a small taste of Planet Money they have an off-shoot show called "The Indicator" which is just a quick 10 minutes on smaller subjects don't make it into the 45 minute main show.
From Planet Money I got hooked on Freakonomics, another economics-based show, but focuses on where psychology/sociology of people meets economics. That's where the classic rational economic concepts break down and things get interesting, because people are not as rational as we like to believe (i.e. behavioral economics). A great episode to start with is #311, which will relate to anyone who has ever bought a live concert ticket (probably most people).
Science Vs. - they explore the science behind many questions and controversies society faces... some serious (like vaccines, fracking, nuclear power, gun control, etc.) and some less serious (like Bigfoot, ghosts, and cures for baldness). The host is pretty funny too... well if you like puns she is funny, if not, you've been warned! The fracking episode (July 27, 2016) is very good and I personally have enough professional experience in that field to know they mostly nailed the science, which gives me confidence in the show's other topics.
The Memory Palace- The host finds interesting little pieces of history and presents them in 10-15 minute storylike explanations. The subjects are things that are generally too small to get into history books, but interesting nonetheless or related to bigger events. It's like diving into a historical footnote for a few minutes and some of topics are very moving. The host is an excellent storyteller and adds a lot of emotion to history that could be very very dry. Try episode #110 (Lost Camels) or #92 (Oil, Water) to start. The episodes are short, no big commitment. (edited for better link to the show)
Reply All - A podcast about the internet, which may sound kind of dumb, but it's actually amazing. I'm a pretty internet savvy person, but they take incredibly deep dives into online social groups, history, and technologies that I thought I understood, but often I was only scratching the surface of. The hosts can be pretty funny too. The current episode, #126, and episode #109 are great examples of the show.
(edited for typos)