Reading a battered old second-hand copy of "Calculus Made Easy" by Silvanus P Thompson as a teenager was the first step on a path that eventually lead to me getting a degree in mathematics. Until then, I always thought calculus and other higher-level maths topics were out of reach for someone like me, but the explanations in that book were so careful and encouraging that it made me realise that anyone could study these topics with just a bit of effort. I believe the latest edition of the book has been updated to replace the slightly handwavy description of "infinitesimals" with a more rigorous limit-based foundation, but I don't know if the book's magic was lost as a result.
As an undergraduate, I purchased a cheap Dover reprint of "Ordinary Differential Equations" by Tenenbaum and Pollard to supplement an ODE course I was taking, mostly because it had lots of exercises in it. However, I found that book hugely inspiring, and it's become one of my favourite maths books ever. Amongst other things, it essentially has a complete course in Newtonian physics in it, just tossed off in the examples! Highly recommended. I eventually went on to study ODEs and PDEs at Masters level, partly as a result of enjoying that book so much.