Oh, I love this question. I need to get my thoughts out and then circle back to read all your stories.
TL;DR: Stratego, Axis & Allies, Canasta, Settlers, and Tigris & Euphrates in that chronological order.
It started early as a way to spend family time together. Sorry!, Four Corners, Trivial Pursuit (my Mom never missed a pink, I stuck to blue and brown), even Life told a story and let you collect things and level up.
At some point we moved toward more strategic games. Obviously I learned Chess, I maintained an aversion to checkers after my grandfather thought the right way to train a five year old was to trounce him mercilessly. Othello was interesting but so fiddly and once we all realized the power of the corners it became awkward to play.
Stratego was a breakthrough point. There was a real space in that game - bluffing, memory, turn zero strategy. And I could honestly beat adults at it. I still love that one.
Then we enter the trashy years. Abalone was an item of interest and took me way too long to realize the game is broken (the only way to win is to not attack, and once everyone realizes this you just stare at each other across the table until someone decides losing is better than staying at the table any longer). But it was brand new to all of us, maybe the first time we really explored a new decision space together. That was special.
Axis & Allies was played in an overnighter format, where we would eat junk food and put on a boring movie in the background and play until 3am. This one really cemented in me how a game can pull you together socially. The memories are real and meaningful, even if the game is sometimes trivial.
Torpedo Run and Fireball Island were so evocative, they introduced me to the imaginative, storytelling side of the hobby. A little gem called Peg Poker entered my early collection where you rolled dice and tried to assemble poker hands with them, then logged them on a tic-tac-toe board. You could be kicked off a space by a higher hand. This was my first time experiencing the joy of analysis paralysis and min-maxing, spending time with a game alone.
Arena. Anyone fall into this? It was a D&D style 1v1 on early AOL which my friends and I played with pen and paper. From this we got OH MY GOD THE ARGUMENTS I swear to this day that die was cocked. It was cocked and he and I aren’t talking anymore, probably for other reasons but this may bake into it somehow. But also my first taste of role playing, spells, and leveling up.
High school gave way to a lot of card games in the hallways. Screaming Yoda, which it took me 30 years to learn was actually called Shithead but I learned it at Church camp, so… And then Canasta, Mao, Big 2… We played a lot of cards and modified all the rules and made up our own 2p variants and multi-deck variants. Then Magic the Gathering which none of us could afford and I always lost to my friends with higher allowances.
Things took a big pivot in 1995 when a friend gifted me Settlers of Catan for my birthday. This was my first eurogame. I’d never seen anything like it. Still have that copy, frayed and worn. I, for one, still love Settlers and will to my dying day.
After that, Puerto Rico, Carcassonne, Blokus, and eventually Tigris & Euphrates which remains my favorite game.
Of the early trio, Settlers remains truly beloved. Puerto Rico I never liked but have so much nostalgia for it, I can’t bear to part with it. Carcassonne I’ve never liked and I hold onto it because I, before I die, need to figure out why everyone else likes it.