I think of myself as a "geometer" so let me try to offer suggestions for books that I take to be entries to geometry at different levels. Let me first mention that geometry is a huge field and I tried to point this out with my "essay" Geometry in Utopia but I have not updated it in quite some time:
What is especially appealing about geometry is that it is so quick starting. One can get to the "research frontier" much more quickly than would be the case for many parts of mathematics.
Some of these books are surveys rather than texts. Once one finds out what areas of geometry one wants to learn more about there are many ways sources to learn more. I want to emphasize that while some of these books are aimed at high school teachers or students that does not mean they can' t be read with profit by graduate students and researchers. Similarly, the books at the graduate school level can also be read by others profitably.
Coxeter, Harold Scott Macdonald, and Samuel L. Greitzer. Geometry revisited. Washington, DC: Mathematical Association of America, 1967.
Craine, T. and R. Rubenstein (eds.) Understanding Geometry for a Changing World, NCTM, Reston, 2009.
Krause, Eugene F. Taxicab geometry: An adventure in non-Euclidean geometry. Courier Dover Publications, 2012.
O'Rourke, J., How To Fold It, Cambridge University Press, NY, 2011
Berger, Marcel, and Lester J. Senechal. Geometry revealed: a Jacob's ladder to modern higher geometry. Springer, 2010.
Hartshorne, Robin. Geometry: Euclid and beyond. Springer, 2000.
Pritchard, Chris, ed. The changing shape of geometry: celebrating a century of geometry and geometry teaching. Cambridge University Press, 2003.
Senechal, Marjorie, and George Fleck. "Shaping Space—A Polyhedral Approach." AMC 10 (1988): 12.
Goodman, Jacob E., and Joseph O'Rourke, eds. Handbook of discrete and computational geometry. CRC press, 2010.
Grunbaum, Branko, Victor Klee, Micha A. Perles, and Geoffrey Colin Shephard. Convex polytopes. New York: Interscience, 1967.
Grünbaum, Branko, and Geoffrey Colin Shephard. Tilings and patterns. Freeman, 1987.