The number 1 personality behind Bourbaki. Even though he was famous for taking the most extreme positions and was widely dismissed as a radical, his vision of mathematics is one that has largely been adopted by almost all mathematicians everywhere. Reading any piece of mathematical work he wrote, it his hard not to feel the respect and passion he felt for mathematics as a subject.
Singlehandedly developed categorical homotopy theory into a full-fledged replacement for the homotopy theory of spaces (Kan complexes, combinatorial homotopy groups, subdivision, $Ex^\infty$, among many other things) as well as a large part of the foundations of homological algebra (Dold-Kan correspondence), category theory (adjoint functors, Kan extensions), and the modern theory of simplicial localization (with Dwyer) among numerous other achievements.
It's said that Kan's breakthrough paper on adjoint functors convinced Eilenberg and Mac Lane that pure category theory was not only a viable mathematical discipline (rather than a language), but also a deep and rich one.