I second This American Life, Reply All, and On the Media.

Some others I really enjoy:

Freakanomics Radio - It's not just for Econ nerds, though it is great if you like economics. I would say the stories are more geared towards sociatal issues than economic ones.

Radio Lab - If you like This American Life, Radiolab is very similar as it is about real people's stories. They also delve into history and science.

Hardcore History with Dan Carlin - Started off as a one-off show where Dan talked about his take on one subject, but it has evolved into a narrative doing deep dives into historical subjects that is discussed no where else. Start with one of his better series like Wrath of Khans and or the WWI one and see what you think.

Common Sense with Dan Carlin - Dan does another show which is more political and he has an interesting take on the politics of the day. He is neither Democrat or Republican and is more libertarian if anything. His theories can be out there at times but he's always entertaining.

99% Invisible - This is a really popular one. It calls itself a show about design but it's really about how humans shape the world. This one grows on you.

Others: Imaginary Worlds, Surprisingly Awesome, Invisibilia, Mystery Show