These are some of the great books I have studied (or used) during my time at university. The list isn't complete and I'm updating it very soon.
Principles of Mathematical Analysis by Walter Rudin
Calculus on Manifolds by Michael Spivak
Complex Analysis by Lars Ahlfors
Linear Algebra Done Right by Sheldon Axler
Topics in Algebra by I. N. Hernstein
Abstract Algebra by David S. Dummit and Richard M. Foote
Elementary Differential Geometry by Andrew Pressley
General Topology by Stephen Willard
Principles and Techniques in Combinatorics by Chuan Chong Chen, Koh Khee Meng
Elementary Methods in Number Theory by Melvyn B. Nathanson
A Classical Introduction to Modern Number Theory by Kenneth Ireland, Michael Rosen