Here is my list, roughly in order of experience:
Super Mario Bros./Legend of Zelda: The first two games I ever played and I was in love immediately.
Mega Man 2: Found this game for $5 at a flea market and fell in love with the series.
Super Mario Bros. 3: Mario 3 took everything I loved about Super Mario Bros. and expanded upon it.
Kirby's Adventure: Oh man, one of my all time favorites. I played and 100%ed this game many times over.
Super Mario World: It goes without saying, it took the Mario 3 formula and further expanded upon it. And introduced yoshi...
Sonic The Hedgehog 2: Great gameplay, great music, a fantastic sequel.
Blast Corp/Star Fox 64: The Nintendo 64 was my first system after the NES.My first N64 games. I played them so much and love them to this day.
Super Mario 64: The pinnacle of 3D platforming, what a fantastic game. When I first got it, I played it for hours on end and the first encounter with the piano scared the crap out of me.
Banjo-Kazooie: This game took everything great about Mario 64 and expanded upon it. While I still love Mario 64, Rare took the Mario 64 blueprints and expanded upon the genre. This game establish my enjoyment of Rareware games.
Mario Kart 64: Had a ton of fun in single player and multiplayer. The best Battle Mode in any Mario Kart, in my opinion.
Legend of Zelda: Ocarina of Time: Like Mario 64, this game brought another classic to 3D
Super Smash Bros: Played this soooo much. Had a great time and enjoyed messing with gameshark codes to play as Metal Mario, Giant DK, Master Hand, and the fighting polygon team. Word of warning, those codes had a side effect of forcing you to play as Mario, no matter what character you chose, unless you wiped the game save.
Yoshi's Island: I was really late to the party, but the wait was worth it. Beautiful graphics, great music, and great gameplay made this game a true gem.
Metroid Fusion: This is the first game I played of the series and made me a huge fan of the series.
Rhythm Heaven Fever: What a great game. This game got me into the series and I now have beaten all of the games. I even imported the physical copy of the 3DS version before it was released in English and made my way through the original GBA release.
And now for the more recent:
Puyo-Puyo Tetris: Always been a big fan of Tetris and Dr. Robotnik's Mean Bean Machine, so this game was a no brainier. We still play this at family gatherings.
Super Smash Bros. Ultimate: My dream came true with getting Banjo and Kazooie in this year, so the rest it just icing. I have reached smash nirvana and I have nothing but best wishes for the characters you desire.
Undertale: This game came out of the blue when someone recommended it to me and I loved it. It was an interesting experience playing Earthbound and Undertale simultaneously.