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Hard to go wrong with Chrono Trigger, a classic JRPG that still holds up imo. Earthbound is another classic that's a must play, it hasn't aged as well but is a great little piece of video game history.
Those are the two big ones that come to mind with emulation for me.
My favorites are The Legend of Zelda: A Link to the Past and Yoshi's Island. I also like the classic Mario games.
The Dragon Quest series if you're into traditional RPGs (it's the franchise that kicked off the genre basically). DQ1-3 have GBC remakes but are a little dated (3 is a fan favorite. 2 not so much). DQ 4-6 have DS remakes with 4 and 5 being the best entry point in my opinion. Art is by the creator of Dragon Ball too so if you like that it's definitely a plus.
Try Super Mario RPG. Fantastic RPG for SNES made by Squaresoft that never got the remake/remaster treatment.
The Legend of Zelda: A Link to the Past, and more recently Link's Awakening. Also the older Pokemon gameboy titles.
So I've never played a single Zelda game. Which one do I start on?
I did play like 2 hours of BoTW on switch and just never got back to it.
A Link to the Past is the one you want to start on. It'll get you right to the heart of what makes the classic Zelda games so good. If that's too retro for you, play Ocarina of Time, you can't go wrong with that masterpiece.
Should I do the SNES version or GBA?
There are exceptions, but for most games, given a choice between SNES and GBA versions, the SNES one is better. Usually sound is severely compressed on the GBA and the visuals are washed out.
Some of the GBA remakes also added in voice tracks, like Mario saying "Just what I needed!" whenever you get a mushroom, or Link doing his grunts and "hii-yahs" whenever you swing the sword. I guess I could see how some people might think those were neat, but I mostly found them annoying, especially given the quality of the GBA audio.
If you're doing emulation, just go SNES. GBA is only worth playing if you're going to play it on an actual GBA.
Super Metroid on SNES. Amazing, great game. My absolute favorite. And if it interests you, there are a lot of good ROM hacks for the game, giving you near infinite replay-ability.
I've finished it a couple of times as a kid, but playing Golden Axe II again (and again) never gets old.
Natsume games on NES have aged well, eg shadow of the ninja, shatter hand, and powerblade. But they're tough games so could be brutal on touch pad controls. Arcon is a cool mix of strategy and action, and had a good NES port.
Play Gunstar Heroes right now.
If you can find an emulator and a device with no input lag whatsoever and an ideal control scheme, Kaboom! on Atari 2600 is addictive.
NES Zelda or the SNES Zelda
Going by ones I play the most, probably Final Fantasies IV, VI, & Tactics, Link to the Past, Sonic 2, the GBA Castlevania games, and some Pokemon Mystery Dungeon.
I recommend Legend of Legaia on PSX. It's an RPG with an interesting fight system. The story isn't anything amazing but the overall package is very satisfying.
Bomberman '94 for PC Engine/TurboGrafx-16. Most people know it as Mega Bomberman for the Sega Genesis/Megadrive, but that version is a remake with worse graphics, sound, and controls. The PCE version was never released outside of Japan despite having 90% of the text written in English, but it is much better and really fun.
I played Pokemon Snap for the first time last week. It's not very long and only has like 60 pokemon but it was a good time while it lasted. Just a chill rail shooter about trying to line up the best (photographic) shots.
If you want something more action oriented then I had a blast playing Cybernator on SNES, it's a 2D mech shooter/platformer. Dynamite Headdy on Genesis is also solid time, it's by Treasure and every level always changes things up a little bit.
The lunar rpgs are pretty good on ps1
Lots of great options already, but I'll throw in WarioWare on GBA since I haven't seen it mentioned yet. Lots of super-simple fast-paced minigames that are nice for a quick playthrough if you're on the go.
Considering the fact that you're using a touchscreen device which isn't great for things like platforming or action RPGs, I recommend checking out more turn-based RPGs. The Final Fantasy 4,5,&6 ports on the GBA are fun and look great, as are the first two Golden Sun games. If you get your hands on a good DS emulator I also recommend Dragon Quests 4-6 on that platform
Gyruss - NES.
Avenging Spirit - Nintendo Gameboy (B&W)
Going to show my age here… Fool’s errand and Dark Castle. Mac games of much nostalgia.
If you like platformers and haven't done it already you should play both Super Mario 3 on NES and Super Mario World on SNES. Still really great to play but also a good history lesson - a lot of staple design concepts were pioneered in those games!
Staying in the platformer genre, the original Rayman on PSX wil always be one of my favorites. Wonderful, unique art and sound design. Starts off very slow but becomes fiendishly difficult as you keep going. Be advised that it was designed with limited lives and some brutal gauntlets between save points. Save scumming with your emulator is probably going to be good for your sanity here.
It's a bit of nostalgia, personal preference, and just overall masterful developers - but Super Mario World is a solid 10/10, easily the best game I've ever played. SMB3 is excellent as well.
For me, they still stand the test of time. I was a huge Sega fanboy back then, but I also loved Nintendo and the games they had. SMW just always blew me away, so every time I play it I think of that time. How advanced it was, how it played so much better than anything I've previously played. It was a serious masterpiece, and IMO still is.
Definitely a must play. Depending on your age, though, it may seem not as great as I'm claiming. I think it still is, but that's from my point of view. I'd really be interested in younger people's opinion that grew up on the Gamecube, Wii, and later console versions of Mario.
Like @R5N said - a lot of staple design concepts were pioneered in those games. Some that now are very commonplace. That's where I'm curious... Is it just normal for some while for us that played them on release they were so perfect and new?