Heh, I could probably go on about the SNES for quite a while. Incidentally, I'm playing the Switch port of Secret of Mana with my kids (the one that still allows for three players, my brothers and I used to have great fun with that game back in the day).
For racing I used to be quite good (worldwide competition) at Super Mario Kart and F-Zero, both of which are very good. Also did Top Gear for a while.
For RPGs, probably Super Mario RPG, Chrono Trigger, and Final Fantasy III (VI) can't be beat. Secret of Mana just superb. Secret of Evermore never did it for me but the third entry, Seiken Densetsu 3 (which was finally ported officially to English in the Switch collection) is excellent as well. Have played some others that were less memorable. I never did beat Earthbound though.
For platformers, Donkey Kong Country 1, 2, and 3, Super Mario World, and Super Mario All-stars (which has the first three Super Mario brothers game done in SNES graphics and the very difficult Lost Levels). Yoshi's Island has more than an honorable mention.
Super Metroid is one of my all-time favorites. Super Castlevania IV is also a great entry in the metroidvania style.
And who could forget The Legend of Zelda: Link to the Past?
As far as memories, I guess my brother and I had a pretty good competition going with time trials on Super Mario Kart. We would battle regularly and trade best times with each other. Balloon Battle was pretty fun as well, with the constant dodging, saving a feather for just the right jump over a wall, and slamming you sibling with a red shell.
Secret of Mana has many fond multi-player memories too. I can remember many shrieks of "heal me! heal me!" in the midst of battle while trying to take down a boss.
Likewise, my brother and I found every single secret exit and alternate path in Super Mario World on our own, with the exception of one (the one where you have to sacrifice Yoshi, I guess we were just too attached to the little guy). This was before we ever had Internet access. We would go back and scour those levels for any secrets we could find and I got to be quite the expert at flying all the way across the stage with the cape. Tubular!