Here is my list.
I tried to make it more practical by supplying links to electronic versions whenever possible.

Rudin: Principles of mathematical analysis

Electronic version: http://libgen.org/get?nametype=orig&md5=0AB81110FEE9FBF5C218F772AB137601

Kostrikin, Manin: Linear Algebra and Geometry

http://libgen.org/get?nametype=orig&md5=796855351C245310B4FAD3C5947C846A

Cartan: Elementary theory of analytic functions of one or several complex variables

French original is available in electronic form:
http://libgen.org/get?nametype=orig&md5=6E9E2B89719AD44D885B6428D8B9D077

I couldn't find the translated book in electronic form,
but Dover offers an inexpensive paperback edition for $9.

Shafarevich: Basic notions of algebra

Electronic version: http://libgen.org/get?nametype=orig&md5=11154CB5CF3714C07D0D20FB3C79D803

Milnor: Topology from the differentiable viewpoint

Electronic version: http://libgen.org/get?nametype=orig&md5=E78E64CD53429CC8DD94D7282E2BDA27

Hatcher: Algebraic topology

Electronic version: http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Helemskii: Lectures and exercises on functional analysis

Electronic version: http://libgen.org/get?nametype=orig&md5=A18C3A9EC500745D563F9D3816892E3B

Milnor: Morse theory

Electronic version: http://libgen.org/get?nametype=orig&md5=ACD9C232FDFD205E937583F301F20058

Serre: A course in arithmetic

Electronic version: http://libgen.org/get?nametype=orig&md5=C00F38F10D80A59AF2A64B3D6D427CFC

Edit: I rearranged the list so that books appear more or less in order of increasing difficulty
and prerequisites of every book precede it in the list.