Just to make it more (or possibly less) challenging:
Ideal 3-game collection pre-2010?
Ideal 3-game collection pre-2000?
Pre-2010 still allows me Pandemic + On the Brink, but I think I’d have to replace Root with Brass: Lancashire (AKA Original Brass), and Skull with 6 Nimmt
Pre-2000: Much more tricky for me, because there are a lot of pre-2000 “classics” that I haven’t played, and I don’t like Catan 
. So… Through the Desert, Crokinole, and… Condottiere?
I asked my husband for his three games last night without giving him the context. He immediately said Terraforming Mars then “Wait, do I get all the expansions?” I said yes so he kept TM then added Gloomhaven and said Frosthaven counts as an expansion. Then he said “This is too hard! I don’t know. There are too many good games,” and never picked a third.
So we overlapped on one from one version of my list. No real surprise there. TM is not quite his version of my Concordia. It is a think-ier game he absolutely loves like I love Concordia. The difference is he really dislikes, possibly hates, my Concordia while I think TM is barely ok, so I’ll play his more often than he’ll play mine.
Pre 2010: Innovation is replaced by Race for the Galaxy, and Pax Pamir 2e is replaced by Through the Ages.
Pre 2000: Still Go, obviously, along with Tigris & Euphrates and Condottiere.
Pre-2000: El Grande, Through the Desert, Tigers and Pots. (Hard to find heavy ones in this period in my collection. I think thats how the 90s roll back then)
2000-2009: King of Siam, Imperial 2030, Chicago Express
Mine were all post-2010. Pre-2010, I’d go with my other choices of Carcassonne, Race for the Galaxy, and … maybe High Society? It doesn’t have the player count for a proper party game, but it’s simple and light enough.
I considered Once Upon a Time, but it really needs the right group.
I love seeing the love for Power Grid. It’s my favorite game. I have all the official expansions (except for that one promo that changes the connection costs on the Germany map). It does two things I really enjoy in games: route building and upgrading.
Similarly for Terraforming Mars, except map building rather than route building. It is also a game my wife enjoyed when she was in her gaming phase.
I don’t have a game calling out to me for my third game. Concordia fills the same niche as Power Grid. La Granja is the euroest euro that ever euro a euro. Hans Teutonica hits the same sweet spots. I’m a roleplayer at heart, so adventure games don’t do as much for me.
Let’s pick something a little off the standard list and go with Thunder Alley, an awesome racing game with twists and turns (and not just to the left). And if it is only one of three options, I might be able to play a full season/campaign.
Pre-2010 I still get The Resistance. And Rallyman, though I didn’t own it then. And… ooh, Ghost Stories.
Pre-2000: 6 Nimmt, and um. Guillotine? Lord of the Fries?
@EnterTheWyvern thank you for reminding me about Iki. I’ve had it for a while and haven’t felt up to trying to parse its rules. But with that endorsement I’m starting to think I need to try again.
My three: Aeon’s End, Village and Nusfjord. Aeon’s End because I’ve played it 20 times in the past few months and I can’t think of a game I’ve played anywhere near that much. Village because it’s so whimsical and because it had such a special role to play in my proposal. Nusfjord because it produces such a rich, tricky economic puzzle out of only a few resources and only 21 turns.
I don’t think I can whittle my games down to just three. I can easily eliminate a number of my games from contention, simply because I know I do not want to play them constantly, but they are fine every now and then (things like King of Tokyo, Betrayal at House on the Hill, 7 Wonders, etc.).
The problem is, like someone else mentioned above, many of my games I have not played enough to know if they really are things I would enjoy forever. I love the games of Inis I have played, for instance, but 3 is a small sample to know if it really is as great as I think it is. I have many other games that fall in the same category like El Grande and Shogun.
I guess if I were guaranteed players who had a shared passion for the games as I do, TI 4th seems like it would provide a lot of entertainment value with all the different alien races and objectives, even more so if I can factor in the upcoming expansion. I have only had one play of the game, and we weren’t able to finish, but it was an amazing experience that I hope to get to the table again someday.
Ghost Stories is pretty much my favorite co-op, even though we lose so much more often than we win. It is always a great challenge, even in the easy mode we play on. It is a game I would like to get better at, so I feel safe putting it on my list.
I am torn on the third. I sort of want to put something easy to teach and play, so I could play with anybody. Alternatively, I want to put something a bit deeper and soloable on the list. So feeling torn between Lords of Vegas with my new, unplayed-as-yet Up expansion, or Arkham Horror TCG. I guess to avoid having two co-ops, I will go with LoV.
Oh, this is nasty… very nasty…
I will limit myself to my collection, which isn’t precisely very big.
I would keep Gloomhaven for the seer amount of hours that it contains, plus the fact that it reminds me so much of my first boardgame love, Hero Quest.
Love Letter would be a strong second, mainly for its convenience, and the simple mechanics that are so easy to teach.
And I think the third one, after long deliveration, would be Werewords. It does fill the party game gap very well, again, it is very easy to teach and I never had a game that was not fun. Plus having the deluxe edition brings a lot of variety to the game if needed.
But I don’t like the starting premise. Not at all. If anything, it is a consolation there are still loads of different ones in this list.
So for the pre 2000 challenge MtG is a strong contender and some of the classic Knizias and yet:
And from 2000 to 2010
Ghost stories and Robo Rally are strong contenders for me too 
Unfortunately, IRL I am cursed with gaming friends who hate both of them 
Slimming down to 3 seems so difficult, because it feels like I’m choosing games that are versatile over games that I love. I’d hate to suddenly find myself rarely playing any games because the ones I have are too heavy as well. So, we’re looking at accessible and versatile…
Codenames:
Quite liked by family and friends alike. Manages to be both challenging and accessible. I’ve yet to meet anyone who dislikes it. One way or another, I know that if I set it up, it will be played over and over again.
Cyclades:
I don’t love Cyclades, but I can’t think of another mappy strategy game which:
The Resistance:
It’s easily the most accessible social deduction game. Werewolf might be simpler, but it’s hard to convince people to play a game where they may be eliminated immediately. There’s also more definite information available, so it feels less messy. No, my friends/family want to spend the whole 45 minutes convincing people they’re not a bastard when they might actually be a secret bastard.
Gah, I look at those 3 and I hate it. You know what, if I’m only going to get 3 games, I’ll get three 2-player games for me and my partner. Give me Memoir 44, Whitehall Mystery, and let’s imagine a world where they keep publishing more Sherlock Holmes Consulting Detective. There… much better.
My technique was “OK, so let’s imagine I can still play any game I want to, any time I wanted to, which three would I still just own in spite of that” and suddenly it became a lot easier.
Me: looks at empty board.
Me: looks at opponent.
Me: “Good game. What should we play next?”
That was something that made complete sense to me after I’d played it a couple of times. At a certain point adding more stones just seemed an exercise in futility, because it didn’t gain or protect territory.
I haven’t tried to narrow this down perfectly (partly because I don’t think I could), but I’m going to say:
I can teach them to anyone. They cover a range of gameplay styles, durations, and player counts. They are all amongst my all-time favourite games. They each have a bunch of expansions (one of them on a fairly ridiculous scale). A limit of three games is pretty brutal, but if these were what I had at my disposal and someone wanted to play something, I think I’d be in good shape.
I meant to ask about this!
I love tiny box/bag games. Is Hive or Hive Pocket something that’s easy to teach and immediately fun for newbies? Looks ideal as a game to carry around.
I carried a home-made set around the world for a year and a half and found it easy to teach to all kinds of people, and popular too.
I’d probably want you to teach this one to me again, given the confusion regarding the original rules and the revised rules. It’s like Carcassonne farm scoring is that regard, come to think of it. ; )