Some of the books,  that I can remember, from my PhD reading list under Peter Petersen at Ucla for doctorate in Riemannian Geometry which I never completed were:  _Characteristic Classes_,by Stasheff, _Morse Theory_, Milnor, _Dimension Theory_, Hurewicz and Wallman, _The Topology of Fibre Bundles_, Steenrod.

We both agreed that Spivak's $5$ volume _Comprehensive Introduction to Differential Geometry_ was quite useful.   There was Galot, Hulin and La Fontaine's _Riemannian Geometry_.

I'll update this if I remember anything else. 

It seems to me that Milnor's _Topology from the Differentiable Viewpoint_ might have been also, and Spivak's _Calculus on Manifolds_ probably wasn't,  but I enjoyed it. 

I also had a copy of his _Riemannian Geometry_ in manuscript form, which is now available in the GTM series.