I cannot recommend On Numbers and Games by John H. Conway enough. He demonstrates the surreal numbers by proof which is a wonderful introduction to proofs. The latter section of the book is highly entertaining.

He also worked on cool stuff in the sphere-packing problem: http://www.amazon.com/Packings-Lattices-Grundlehren-mathemat...

I really appreciate is writing and teaching style. He likes to bring his readers and listeners up to his level.

Shameless plug: I implemented Conway's Life in Prometheus (the monitoring system) earlier this week: http://www.robustperception.io/conways-life-in-prometheus/

Demo: http://demo.robustperception.io:9090/consoles/life.html

I implemented Conway's Life in K: http://johnearnest.github.io/ok/ike/ike.html?gist=48e0693b18...