Here goes nothing. I have many other suggestions that are not listed; there is always one more example of a great math book:
"The enjoyment of Mathematics" H. Rademacher, O. Toeplitz. You can't beat this classic collection of serious (and beautiful) results with elementary proofs.
"A mathematician's miscellany" J. Littlewood. This is much more advanced, but it is a wonderful collection of math, anecdotes, and insights into the life of a professional mathematician. Sometimes it has a very pedantic tone, but always, always profoundly human.
"Development of mathematics in the 19th century" F. Klein. You can browse and browse and always extract a bit more insight into what those mythical giants were thinking about when they created modern mathematics.
Anything by M. Gardner. Our enterprise is a serious one, but its recreational side is a major source of inspiration and ideas. The older books are better in the sense that they portray better the network of curious minds that kept in contact through Gardner in the primitive pre-internet ages.
"Goedel, Escher and Bach" D. Hofstadter plus anything by R. Smullyan. Someone said that Goedel's Theorem is the one result from 20th century Mathematics that every mathematician should know.
Anything by Milnor. I have said it elsewhere. If you want to learn to write better, absorb the style of the masters.
"Three-dimensional geometry and topology vol. 1" W. Thurston. Just open at random and swim into an ocean of ideas. The original Thurston notes are already available in the web. They are very different from the book, so you can swim into two connected oceans.
"Complex analysis: an introduction" L. Ahlfors. Although the language feels now a little old, you can open almost any page and find jewels of exposition. I am always surprised by the incredible precision in the choice of proofs, exercises, topics, and topic order. It is an incredibly methodic book and very thorough.
"Differential equations, dynamical systems and linear algebra" M. Hirsch, S. Smale. I was told to study by hunting for the many, many, typos. The newest edition has been cleaned, updated, and expanded by Devaney, but it remains a great book.
"Ordinary differential equations" V. Arnold. Even if this is not your thing, you should read at least the first three chapters.
"Introduction to commutative algebra" M. Atiyah, I. MacDonald. A short book, and very readable (at least the first few chapters).