I wasn’t aware of the aural joke in Troilus and Cressida:
In the mid-1590’s, a man by the name of John Harington developed (or at least is credited with developing) the first flush toilet in Britain. Until this time, toilets were known by the slang term “jakes,” so it is no surprise that Harington called his invention the “Ajax”… punning on the idea of this being “a jakes” … a toilet.
A piece of trivia: it’s thought that the use of the word “john” to refer to a toilet is also a reference to Harington.
So for Shakespeare to have a character named Ajax in a play about the Trojan war may have been no surprise, but to make him a character so full of crap, so to speak, makes for quite the joke.
Another piece of trivia: Kit Harington, late (? spoiler? maybe?) of Game of Thrones, is supposedly a descendant of the inventor of the toilet… “You know nothing, Jon Snow–about toilets” (see what I did there?).