If I had to only choose 10 games to keep in my collection, and want to cover as wide a swath of game types as possible, here are the ones I would keep:
Codenames
Simple, easy-going party game with enough teeth to it that it stays interesting, but relaxing enough that you don't have to get too into it if people just want to relax. Might replace this with Anomia depending on the day of the week.
Time's Up
Extremely tense, competitive and exciting charades-alike. I've played this at every family gathering for the past 8 years and it never gets old. Possibly my favourite game of all time.
Skull
My favourite bluffing game, full of prisoner's dilemmas.
One Night Ultimate Werewolf
My favourite hidden role game. The Resistance is more elegant, but ONUW wins out for punchiness and zaniness.
Modern Art
Now we get into the heavier stuff. Modern Art is the best auction game I've played, which makes sense because it's got every kind of auction in it. It's the epitome of the sort of game where you finish and think "Oh, that's what the strategy is"
Brass: Birmingham
I've never played the original Brass, but I don't think I need to, because Birmingham is so good. It's a tightly-wound spring of a resource-management Euro game, where every decision tugs on every other one. A bit overwhelming, but very satisfying.
Concordia
Concordia's like a much looser, much gentler cousin to Brass. Where a bad play in Brass narrows your options to a pinprick, every choice in Concordia sprawls into a web of other possibility choices, each drowning you in a sea of resources and points. The joy is in finding the most efficient path.
Bohnanza
My favourite negotiation game. Quite brutal depending on the group, but extremely raucous and fun.
Battle Line
Hard to say whether this or Modern Art is my favourite Reiner Knizia game. Great card game, with chess-like decision making and a bit of bluffing thrown in. My kingdom for a new edition with better card art.
Jaipur
A nicer, more easy going two player game. Easy to just play over a conversation and quite nice looking.
Inis
Drop-dead gorgeous and the most dynamic area control game I've played. Endlessly fun and interesting. Plays great with 2, 3 or 4.
Honourable mentions: Twilight Struggle, Star Wars: Rebellion and War of the Ring. I don't own any of these, but only because the guy I play 2-player war games with already owns them. If you have somebody to play big two-player war games with, these are all great.
As you can probably tell from the list, my interest tends toward the hobbiest side, but these still cover a pretty big swath of types of games. This list assumes that I am keeping these games around because I am a person who likes to set aside whole afternoons to play games (because I am that person). I have played every single one of these with siblings, parents and cousins, though, and every one of them I've played at least a dozen times I think. (well Brass I've only played with my brother, and he's into heavier stuff, but still)