What is the single most influential book every mathematician should read? If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
A Mathematician's Apology by G H Hardy. I did in fact read this in high school, and it raised my view of mathematics from a thing of utility to a thing of beauty and wonder. It inspired me to go on to study mathematics at Cambridge myself.

It's a pity that the "introduction" by C P Snow is longer than the original and contains a rather depressing view of Hardy's later life. I would recommend readers to skip the introduction altogether and concentrate on Hardy's own words.
I would read a book about Perelman's proof of the Poincaré conjecture (or even the papers themselves). Oh, you mean the book had to be written when I was starting?