I tried this post on StackExchange with no luck. Hopefully the experts at MathOverflow can help.
In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses.
They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead Elements of Homotopy Theory. These are both excellent books that (theoretically) give you overviews and introduction to most of the main topics that you need for becoming a modern researcher in algebraic topology.
Differential Geometry seems replete with excellent introductory textbooks. From Lee to do Carmo to so many others.
Now you might be thinking that Kobayashi/Nomizu seems natural. But the age of those books is showing in terms of what people are really doing today compared to what you learn from using those books. They just aren't the most efficient way to learn modern differential geometry (or so I've heard).
I am looking for a book that covers topics like Characteristic Classes, Index Theory, the analytic side of manifold theory, Lie groups, Hodge theory, Kahler manifolds and complex geometry, symplectic and Poisson geometry, Riemmanian Geometry and geometric analysis, and perhaps some relations to algebraic geometry and mathematical physics. But none of these topics completely, just as Switzer does with a unifying perspective and proofs of legitimate results done at an advanced level, but really as an introduction to each of the topics (Switzer does this with K-theory, spectral sequences, cohomology operations, Spectra...).
The only book I have found that is sort of along these lines is Nicolaescu's Lectures on the Geometry of Manifolds, but this book misses many topics.
This was inspired by page viii of Lee's excellent book: link where he lists some of these other topics and almost implies that they would take another volume. I'm wondering whether that advanced volume exists.
Any recommendations for great textbooks/monographs would be much appreciated!
Edit: there are many excellent recommendations (I particularly like the Index theory text mentioned by Gordon Craig in the comments as it doesn't shy away from analysis, and does so many things in geometry plus has extensive references) below. One other reference that I found which people may find interesting is the following: link and link2 where Prof. Greene and Yau say: "It is our hope that the three volumes of these proceedings, taken as a whole, will provide a broad overview of geometry and its relationship to mathematics in toto, with one obvious exception; the geometry of complex manifolds...Thus the reader seeking a complete view of geometry would do well to add the second volume on complex geometry from the 1989 Proceedings to the present three volumes". However most of the articles are research level articles and lack the coherence and unified vision of a textbook/monograph.