[calstad]

These fall more in the "pop-sci" realm, but I think are really good at trying to convey what kinds of things working mathematicians actually think about:

- Colossal Book of Mathematics by Martin Gardner: Uses games and recreational math to introduce areas like group theory and topology.

- Finding Moonshine by Marcus Du Sautoy: Tells the story of the unexpected connection between two areas of math that, on the surface, seem to be completely unrelated.

- The Great Mathematical Problems by Ian Stewart: Does a good job explaining several of the major open problems in math today.

[eimrine]

Ian Stewart never disappoints and seems related to OP's request. His "Nature of Numbers" is exactly that, also this is the first English book (foreign language for me) which I have read without any need of dictionary.

"The Great Mathematical Problems" is about reading a chapter, then taking a rest, then reading another. "Nature of Numbers" can be read in one attempt and my emotion about that is I want this book to be several times thicker. The illustrations are among the best mathematical images I ever seen, despite being monochrome.