Hitchhiker’s Guide to the Galaxy. Planetfall. Stationfall. Bureaucracy. Spellcasting 101. Leisure Suit Larry.
Space Quest starring Roger Wilco.
King’s Quest!
I agree with the Fallout games, especially the second; lots of dark humour and pop culture references. Black Isle are good at that sort of humour; Planescape: Torment is a dark, very philosophical kind of game, but it has some side-splitting humour too (in a weird kind of way, admittedly).
There were also the point and click games based on Terry Pratchett’s Discworld. I found the first one quite funny, but the second one’s jokes got much further removed from Pterry’s kind of humour, and used too many Monty Python gags instead of original stuff (I hate it when ripped off jokes are supposed to be funny just because the humourists Ironically Recognize that they’re ripping them off)
Baldur’s Gate II had quite a bit of humor to it - mostly from Minsc and Boo.
One of my favorite parts of it was a very obscure reference to an internet oddball. There’s this guy who has catalogued every appearance of squirrels in role-playing games, and has a webpage about this. Almost always it’s just a squirrel seen in background animations. Anyway, there’s this part in Baldur’s Gate II where your party is wandering through the wilderness and suddenly a couple of squirrels show up, running in circles, and Minsc says ‘Look Boo! Squirrels!’. This was supposedly a reference to that site.
MLB Slugfest has the funniest commentary of any game ever. I don’t even like basball games but I like that one only because it makes me laugh, HARD.
Fallout 2lot of funny moments.
I thought the diahrreah chapter of PaRappa was pretty funny.
All these great Lucasarts games, and no one’s mentioned Sam & Max Hit the Road? That game still cracks me up to this day… it was directed by Tim Schaefer, the same guy who did Day of the Tentacle, Full Throttle, and Grim Fandango. Too bad he left Lucasarts a few years back…
Incidentally, I completely agree with what others have said about Fallout 2–there’s some hilarious political satire going on in that game. My favorite example is the conversation with “Vice-President Bird,” who speaks only in Dan Quayle quotes, enabling you to choose the dialogue response “You’ve lost your mind.” Great stuff.
The new Zelda game, The Wind Waker, has quite a few very funny parts… such as Link being shot out of a cannon in a barrel in order to get inside an enemy fortress. Simlarly, the Megaman Legends series is great–awesome games, with funny characters and a lot of very entertaining moments.
Come to think of it, Kingdom Hearts has some great comedic parts, too–especially if you’re familiar with the Final Fantasy and Disney characters who show up in it.
I found Monkey Island 2 hilarious … but I thought the humor in the later ones was very forced.
Funniest game ever : The Neverhood.
Imagine a 20-year old Salvador Dali making a video game with a bunch of his buddies, 100 crates of beer and 30 tons of clay.
Be sure to watch “The Making Of …” video on the CDROM.
I never got round to Grim Fandago … I’m gonna have to try to track it down and try it.
I find Dungeon Keeper 2 to be quite funny at times, especially with the really deadpan announcer and lines like: Your minions demand cable.
I second the Lucas Arts nominations. The cartoon-quality and humor of those games really appeals to me.
I’ve never heard of “Neverhood,” but am going to look it up based on kevlaw’s recommendation. 
Conker’s Bad Fur Day on the N64 actually had some quality humour. Surprised the crap out of me, I don’t laugh at anything.
Most of it is lame toilet humour but occasionally something honestly funny pops up.
“Darkened Skye” looked pretty damn funny when they reviewed it on “Extended Play”.
Hey, shut up, man, Phantasmagoria was great!
I liked “Neverhood”, too. The music was funny, as well.
“Zork: Grand Inquisitor” had a bit of humor you may not have recognized. The picture of Belboz the wizard (who doesn’t have an active role in the game) is Stan Freberg, the great satirist and father of the funny commercial. Stan isn’t listed in the game credits. Also, in the wizard’s home, one of the messages on his answering machine is clearly Donavan Freberg, Stan’s son, who is in the credits. (Donavan was the blonde kid on the Encyclopedia Britannica commercials from years ago, and Stan was the announcer.) The adventurer comments on the phone message saying “Isn’t that the kid who was on those Encyclopedia Frobozzica commercials?” 
“Eric the Unready”, which was basically a funny text adventure with still images added. Slapstick and Pythonish humor, and references to “Blazing Saddles” and “Star Trek”.
I started playing “Neverhood” but it was very, very strange. I can’t remember talking to anyone in that game or anything particularly funny - the only thing I remember was this wall of legends which went on for infinity, or at least a really long time. Should I try to dig this game up again? I LOVED the Monkey Island games - they’re hilarious. Speaking of which, does anyone know how I can get Monkey Island II? LucasArts doesn’t sell it anymore because it’s so old. Amazon doesn’t either, though they have a Monkey Island Archives thing which has unfortunately been discontinued.
I found that TimeSplitters 2 had a few funny moments in the actual gameplay (Tripper’s mirrored shades reflecting the “Goldfinger” laser), but most of the humor comes from the characters. Most of the enemies in one level wear goofy yellow jumpsuits, except for the elite few who wear black. The background information for the black-suited ones says that Dr. Khallos had some black jumpsuits lying around before buying yellow in bulk, and these guys had first choice.
My favorite is R-107, a poor schmuck who thinks he’s a robotic killing machine. He’s a guy wearing a paper bag over his head and a cardboard box with the word “robot” written on it held on by suspenders. He also tries to sound like a robot whenever he gets shot - “Bzzt!” “Stop that!” and “Why?!” in a silly pseudo-electric voice.
I agree; game makers don’t seem to be attempting to make many funny games any more. One recent example I can think of, though, is Outlaw Golf, for the Xbox. Between the genuinely funny announcer, amusing character and course design (one of the courses is underneath/around the New Jersey Turnpike), and the fact that you get to beat up your caddy to alleve pressure and calm yourself down… this game had me laughing out loud a number of times.
Another funny old LucasArts game that no one has mentioned yet, probably because it was never particularly popular, was Zombies Ate My Neighbors, for the SNES. I loved that game!
I mentioned it, in the OP. And Schafer didn’t have anything to do with Sam & Max Hit the Road. It was written by Steve Purcell (the creator of Sam & Max) and Colette Michaud, with Mike Stemmle and Sean Clark.
Oh, so you did. Oops. That’s what I get for skimming. By way of restitution, I’ll mention the SNES shooter series Parodius–as in “PAROdy of GraDIUS.” Some very wacky, utterly insane stuff in those games. My personal favorite is Goujuko Parodius (the second one)–which, among other things, features a level centering around a battle with a heavily-armed submarine that has a cat head and legs, all to a version of “Stars and Stripes Forever” being “meowed” by cats. Lunacy, you say? Well, yeah… but funny!
Even though they were never released in the US, the Parodius games aren’t too tough to find ROMs for. Of course, I wouldn’t know anything about that…
A lot of the cut scenes in Ratchet & Clank were pretty funny.