I remember that at one time there was a thread about how many typefaces a designer really needs.
I just came across one famous list of a very few typefaces - by the noted designer Massimo Vignelli.
His list was: Garamond, Bodoni, Century Expanded, Futura, Times Roman and Helvetica.
In my opinion, that is an excellent list. These six typefaces are all very popular, and they're popular for good reason. (Even if Times Roman and Helvetica are so overused that they will draw some negativity!)
Of course, though, it's also easy to think of a second-string list of typefaces that are still also very worthwhile; for example, Centaur, Bembo, Baskerville, Caledonia, and Gill Sans. (Although, personally, I don't like Gill Sans.)