7 Wonders and Dominion are the games that really got me serious about game design. Dominion should probably already be on there, but 7 Wonders should too.
Yeah, I think that both of those definitely belong.
Chess! And Puerto Rico.
Puerto Rico for sure, but Chess is a game that lots of people have played anyways, so I think for a 2-player pure strategy game I'll go with Lympi's suggestion: Hive.
Possibly something like Fluxx, not necessarily for how good or bad it is but for demonstrating how you can have a simple set of rules but interesting emergent behaviour from interactions of components.
On a similar note, I think it's good for a game designer to be familiar with some of the "classics" like Snakes & Ladders and Monopoly, to understand what "real gamers" dislike about them and why (in particular, things like the absence of meaningful decisions, runaway leaders, long end-games and player elimination).
I haven't played Fluxx, but I think that for a party game I'd prefer
Eat Poop You Cat. For simple rules yielding emergent behaviour, I actually think a good game is
Blokus. All you do is place tiles such that you touch your own only diagonally, and the rest is emergent behaviour.
Regarding "classic" games, I think I'll put a note on the list that they're good examples of games not to emulate, but I would never recommend that someone subject themselves to any of those.
I want to recommend Bang and probably some regular card games like Hearts, Spades, Crazy Eights, Poker...
Carcassonne for tile laying?
Settlers and Ticket to Ride, as two extremely popular and successful games?
Bang's a good addition and also a good example of poorly written rules. Every time I play that game I end up having a debate over the rules. I think I'll also put Hearts on as well, as a "regular" card game. Carcasonne is also a good call.
Pretty much everyone's played Settlers, so I don't feel any obligation to recommend that people play it, but Ticket to Ride I think I'll definitely put on.
In order of increasing complexity…
Hive is a brilliant abstract that's simultaneously simple and extremely complex.
Lost Cities is a nearly themeless card game with a surprising amount of depth for a game that is so easy to learn.
Bohnanza is also simple with some very unique mechanics.
Pandemic popularized cooperative games and is definitely worth studying, I think.
Finally, I haven't played Eclipse, but just by looking at photos and videos online, I can tell the way its designed visually and functionally, is super well-thought out. It conveys complex information in a simple manner. In the hands of a lesser designer, it could have been far more fiddly than it is.
Hive is a good addition, and Lost Cities I haven't played but from what I here it's also a good addition. Bohnanza and Pandemic I also agree with, and Eclipse I think I'll put on, but I'll also make note that it's a pretty heavy game. And I have to say that there's one part that's not as super-well thought out as the rest: the combat system. It's better than Risk's system, but it's pretty ridiculous and often quite frustrating.
Regarding Munchkin vs. Gauntlet of Fools, I've only played about half a game of Munchkin and haven't played GoF. However, I have to agree with Ozle, I do somewhat like the politics of Munchkin. I think for now I won't put either on the list, unless someone identifies an important game mechanic that one of them has, but the list is missing.
List so far:
Dominion
7 Wonders
Puerto Rico
Eat Poop You Cat
Blokus
Bang
Carcasonne
Ticket to Ride
Hive
Lost Cities
Bohnanza
Pandemic
Eclipse
