PixlBit

I'm definitely enjoying Fire Emblem Engage. It's story is rather goofy (I do like FemAlear's design, I will admit), but it does take me back to the Tellius games, Path of Radiance and Radiant Dawn, which were my jumping-on point with the series. 

Stage Select:

Playing several of the Zelda games again, namely the first game, A Link to the Past, Twilight Princess, and Breath of the Wild, which are my favorites. One of the things I loved about Breath of the Wild is that it did kind of remind me of my first time playing the original game, at the age of 10, back in 1987.

Ultima III: Exodus was probably my first "traditional" RPG. I've since beaten it and know how to blow through it, but back in 1988, it seemed so big, mysterious, and full of unlimited potential.  The same thing could be said about Starflight, but again, I got a sense of that recently when I started playing No Man's Sky, which has quickly become a firm favorite.

Final Fantasy VI, my first Final Fantasy, and FFVII, which had an unrivaled sense of spectacle at the time.

Dragon Quest VIII, which was special because I played it with my wife the first year I was married.

Cage Match:

I had to go to Google to look these games up, since I could only vaguely remember these names. I'm going to give it to Dr. Muto out of respect to Ed Logg, who is a gaming legend. 

STAGE SELECT:
I know it's not really the question but I'm going to be the twit that says, "None, really." There's so many neat games out there and I want to play them all, damn it. Damn you, time! There are a bunch of games, like any early Super Mario Bros games, that I just can't play without having such a subjective stain on things it's practically solipsistic. Weirdly it's not every game I played in childhood, I replayed Metroid and Zelda 1 and they're fine, experience things I don't remember and the games, while not exactly fresh, still take some getting used to. Not the same for Super Metroid and most of Zelda 2 though! I can practically play those in my sleep. Once I also got bad feedback when I recommended Mystical Ninja for the SNES, it didn't even occur to me how harsh the overworld segments can be. Just didn't even register!

Oh, also a lot of adventure games would be good ones to replay fresh, or anything you need to figure out. I could mention the classics (and I'll mention some real adventure classics like The Maze of Galious and Legacy of the Wizard), but for the other part I'll say that Fez is such an experience that really you can't go back to. Once you know everything, the puzzling is still interesting but it's nothing like the first time. A lot of games where the gimmick is figuring out the controls, like Umihara Kawase or Getting Over It with Bennett Foddy, are games you can certainly keep trying to master but, in a way really, it's not the same game as the one you play while struggling.

Finally, if nobody mentioned it I'll say any immersive sim. Part of the fun in Hitman is experimenting with things but that becomes more limited when you master it (then it's just weird edge cases and stealth stuff). It's less interesting when you realize 90% of items are just the same throwable melee attack. Although I want to be clear, I'm not smacktalking Hitman. To that game's credit, it is immersive sim-y enough there's some really weird edge cases and strategies to master (and because of that it has an insane speedrun/challenge community). But I figure it's another good example of a game where the meta changes drastically once you get used to it.

CAGE MATCH:
Dr. Muto's got Vexx on the ropes! Who will save the Astara from the Shadowraiths now? But what's this!? By gawd, it's Iggy's Reckin' Balls to the back of the head! Wow! They're gonna have to finish the Genitor 9000 just to rebuild his skull! What a duo, definitely our favorite tag team (checks card), Vexx and Iggy's Reckin' Balls! Vexx wins!