Roxanne might be Steve Martin’s best comedy. I have no idea how well some of his earlier films, like The Jerk and All of Me, hold up.
Gremlins 2 is more of a comedy, and more fun IMHO, than the original.
I was noticing there was no Mel Brooks on the “So far” list, and only two on “Coming up.” This seemed wrong.
You got the best of the ZAZ gag-driven comedies, but I also recommend their more traditional, plot-and-character comedy Ruthless People. And Jerry Zucker’s Rat Race.
Haven’t seen anyone suggest The Princess Bride yet, so I’ll toss that in.
Seconding the Marx Brothers (duh!), particularly Duck Soup (also duh!) and Monkey Business.
Despite having already been a dyed-in-the-wool Marx Brothers fan by the early 1970s, Monty Python and the Holy Grail is still the funniest movie I’ve ever seen. So seconding that.
Recommendable - if not precisely “classic” - foreign comedies:
Up to his Ears (1965) – Wacky and (at times) cartoon-like adventure comedy
The Brain (1969) – Heist comedy
Le Magnifique (1973) – Spy spoof
Peking Opera Blues (1986) – Period female-bonding action-comedy with spectacular climax
Project A, 2 (1987) – Jackie Chan is cinema’s greatest practitioner of physical comedy (NOTE: it is not necessary to have seen the inferior first film to appreciate this one)