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.
