In what follows I’ve restricted myself to those books I actually used at the time (nothing from after about 1991 or 1992), so this does not include anything that appeared afterwards.
I had a one-semester course out of Royden (covered most of the book --- it was a fairly fast-paced course), a one-semester course out of Taylor’s General Theory of Functions and Integration (covered the middle third of the book, this being at a different university), and a two-semester course using Wheeden/Zygmund's Measure and Integral that was taught by Torchinsky (6 years before his 1988 book appeared, but much of what's in his book was included, in particular all the stuff on cardinal and ordinal numbers and order types). But these are introductory texts that Ph.D. qualifying exams are based on, and of course I and pretty much everyone else was also familiar with several other books, in my case probably the most significant were (in order of how much I studied the book) Measure Theory by Halmos, Real and Abstract Analysis by Hewitt/Stromberg, Measure Theory by Cohn, and Integration by McShane. Also, Real and Complex Analysis by Rudin (first third of the book) was often recommended, and many students used it for Ph.D. qualifying exam preparation, but for whatever reason I never looked at it all that much, and in fact it was only 4 or 5 years ago that I finally wound up getting a copy of Rudin’s book (saw it at a used bookstore).
For what it’s worth, a couple of books that I recall being recommended very often were (this being late 1970s to early 1980s) Functional Analysis by Riesz/Sz.-Nagy (first third of the book, the three chapters covering integration) and Linear Operators. Part I. General Theory by Dunford/Schwartz (also first third of the book, which is also the three chapters covering integration), but I never wound up looking at them very much. However, in looking at these two books right now, I believe the first third of Riesz/Sz.-Nagy would make for a good summer project if someone wants a refresher in classical analysis topics. But not Dunford/Schwartz, unless you really want to dig deep into set function machinery stuff. Another book (2 volumes, actually) that was often suggested is Theory of Functions of a Real Variable by Natanson, which incidentally I later wound up using often as a reference, but because everything is restricted to the real line the drawback is that you’re not going to see any general measure theory.
What you should read AFTER the basic first year course material will depend hugely on what area of analysis you intend to work in. In my particular case, and this is a rather outlying area (but which I find fascinating), the books that I wound up making the most use of include Real Functions by Goffman, A Primer of Real Functions by Boas (1981 edition), A Second Course on Real Functions by van Rooij/Schikhof, Measure and Category by Oxtoby, The Geometry of Fractal Sets by Falconer, Fractal Geometry by Falconer, Differentiation of Real Functions by Bruckner (1978 edition), Real Functions by Thomson, and Theory of the Integral by Saks.