Well, basically all of my favorite games
I'm a point-n'-click adventure person, so I will start with:
Monkey Island Series: Yes, it has a small cult filing, but my argument is that it was wrongly ignored by LucasArts, when they decided to stop making humorous adventures and start making Star Wars Everything.
The Longest Journey: One of my top games. Its got a great story, challenging puzzles, but not annoying ones, a lot of good humor, beautiful soundtrack, and it basically deserves a lot more attention than it gets. AND it's on Steam for $10 (or maybe $20). So buy it! Buy!
(by the way sequel (Dreamfall: The Longest Journey) is different, good story, not bad gameplay, I reccomend it if you want a good story, but play after the first one.)
Grim Fandango: I haven't gotten a chance to play it through all the way because of technical difficulties
Right now it's being ignored by developers who could easily make a selling re-make or sequel that at least I would buy
(Okay, that's a selfish one
)
Beyond Good and Evil: It should have not taken the developers this long to make a sequel! Constantly on the underrated games list, it had fun gameplay, a great soundtrack, and nearly everyone who I've talked to who played it love it! Play it.