A few suggestions of books that I have either read or read parts of:
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The World of Mathematics edited by J. Newman. Wonderful collection of all sorts of cool math texts, including texts from really famous mathematicians like Descartes and Russell.
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How to Bake Pi by E. Cheng. Interesting and very accessible book where math and category is compared to baking.
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The Mathematical Experience by P. David, R. Hesh and E. A. Marchisotto. Another great (and slingtly more down-to-earth) book about the nature of mathematics.
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Mathematics by T. Gowers. Short guide that tries to introduce the reader to what higher mathematics really is about.
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How to Solve It by G. Pólya. The book about mathematical problem solving.
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Art and Craft of Problem Solving by P. Zeitz. A great book about basic problem solving techniques with a lot of really fun problems.
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Surely You're Joking, Mr. Feynman! by R. Leighton and R. Feynman. Stricly not math, but still a good read!