Read until you drop or solve a problem. Repeat indefinately.
-1. Any time and any place
The Moscow Puzzles, by Boris A. Kordemsky, the most fun and varied collection of mathematical games, riddles and puzzles suitable at any age.
Experiments in Topology, by Stephen Barr, has you cutting and pasting funny spaces, experimenting with Euler's Formula, the Klein Bottle and Projective Space. After this book you'll see symmetries where you would usually get your eyes twisted, and know how to get on in a neighborhood, even though by now you'll be eagerly coloring nonsensical maps.
0. In a probability cloud, condensing possibly right before college
Geometry and Imagination, by Hilbert and Cohn-Vossen, the single one most beautiful text on Geometry. If you were to take a book as a cast out on a lonely island, this book would have you all set. I recommend reading it any time between the ages 15 and 150, whether you do math as recreation, as an undergrad or as a professional.
1. Undergraduate Mathematics
- Proofs from THE BOOK, by Aigner and Ziegler, a remarkable collection of proofs and mathematics. This companion will safeguard you against boredom throughout your undergrauate studies, and complement them to make for a well rounded general knowledge.
- Geometries, by Alexei Sossinsky, spans the bridge from classical to modern geometry backwards.
- How surfaces intersect in space, by Scott Carter. Surfaces and curves are the easiest types of spaces, maybe. Find out how they sit within another space and with respect to each other. Plenty of beautiful pictures.
- Topology of Numbers, by Allan Hatcher, a funky geometric introduction to number theory, accessible very early on.
- Galois Theory, by Miles Reid, develops Galois Theory in an organic intuitive way.
- Frobenius Algebras and 2D Topological Quantum Field Theories, by Joachim Kock, especially if you like surfaces, then this is a well motivated, friendly and very clean introduction to category theory.
- Differential Geometry in the Large, by Heinz Hopf, asks classical questions about curves and surfaces and develops the subject of differential geometry from first principles guided by them.
More to come...