I'm surprised that nobody's mentioned almost anything by Emil Artin. His little monographs on Galois Theory and the Gamma Function are thrilling to read. They are so clear, and use the minimum necessary (but not more -- to paraphrase Einstein) I found them inspiring. Also his "Algebraic Numbers and Algebraic Functions" and "Geometric Algebra".

Another book, is G. H. Hardy's "Pure Mathematics". That's the book that I really learned analysis from (when I told that to Pat Gallagher he exclaimed that I was really lucky) when I was in high school. Reading that cemented my feeling that I wanted to be a mathematician.