Actually a book that ended up changing my life: Kaczynski, Mischaikow, Mrozek: Computational Homology
I read it while working on my master's thesis in computational homological algebra, in order to see what they had to say about efficient implementations of simplicial homology.
After reading it, I first realized that algebraic topology has applications far outside what I had seen thus far - and now, a doctorate later, I'm active in the field of Applied Algebraic Topology and Topological Data Analysis.
I'm not certain I'd peg it for great writing as such, but the criteria above did state "changed my life".