I played through the Golden Sun series and loved the story, lore and battle design.
Dragon Quest as a series has always been interesting, I even loved the old Dragon Warrior games on NES.
Zelda games...so many good games, and my top three are Wind Waker, Majora's Mask and Minish Cap in no particular order.
Super Mario World, RPG and Paper Mario TTYD are, imo, the best ever in that collective.
Sonic and Sonic 2 were among the first games I got for my birthday as a kid.
I'm not surprised nobody has mentioned Contra yet, but I am still playing that. I'm trying for a "no Konami code" run, getting stuck on Stage 4.
Pokemon Mystery Dungeon was the game that gave me what I wanted from the series: playing as a pokemon.
Crash Bandicoot and Spyro the Dragon were epic childhood times. I especially loved CTR, Crash 3 and Spyro 3.
Banjo Kazooie, Banjo Tooie, Donkey Kong 64 and 007 Goldeneye were so amazing...
Chrono Trigger was my first official foray into JRPGs, and still the most memorable.
Earthbound was something I caught onto much later than I should have.
Fire Emblem still remains one of the most epic tactical RPGs I've played.
Having said that, Final Fantasy Tactics is so rich and interesting to play.
Final Fantasy still has some of the most memorable mechanics and interesting stories.
Metroid--specifically Super, Fusion, Zero Mission and Prime--are games I replay to this day.
I was introduced to Kingdom Hearts, Ratchet & Clank, Jak and Daxter and Guitar Hero by the same friend, and I love them all so much.
I only recently started into the Tales games with Symphonia, but I love it so much.
Gradius, Darius, Blazing Star, Life Force, a multitude of forgotten names in the same line of epic space shooters from my youth.
Smash Bros. was the second fighting game with mechanics I mastered (the first being Tekken).
Megaman (PS2 collection), Megaman X (PS2 collection), Megaman Zero (collection on DS), Megaman Battle Network (still have 3-6)...
Portal and Portal 2 are some of the best examples of "story while playing" games, and a non-combat first-person game at that.
Dark Souls and Bloodborne are so much fun and so frustrating, it's the kind of game I'd buy new controllers to keep playing.
I've spent years playing League of Legends, Counter-Strike and whichever MMO my friends were into at the time.
That's...not even a fully exhaustive list, and I still have so much more. I mean, my Steam library is nearly 600 games as of writing this, and I've played maybe a tenth of it thoroughly, maybe a sixth of it enough to form a good opinion of several games. I'm just going to stop here, because if I were to list all the Steam games important to me, I'd probably be late to work. You can probably tell what my latest favorite is, anyway.