First system I saved up to buy on my own, though in fairness it probably wasn't much "saving" due to birthday and Christmas money.

I'm not sure exactly the order of things, but my first games in some order were Super Mario World (packaged in), Final Fantasy IV (which I believe was the first non-computer RPG I bought), Gradius III, Thunder Spirits, and SmartBall (the latter three all being gifts).
Still my second favorite system, and with Virtual Console and certain GBA/DS releaes helping me find stuff that I couldn't afford or that wasn't released for English audiences back then, my opinion of the SNES has only kept improving to this day. (Prime examples of the latter would be Final Fantasy V Advance and Dragon Quest V and VI DS.) Top picks for games would be:
Chrono Trigger - Yep.
Final Fantasy VI - This was the FF that struck the best balance for me in terms of not letting every character be able to do everything (FFV, for example) while also not being too simplistic (FFIV). FFIX also handled its balance well, but unfortunately suffered ludicrous loading times.
Kirby Super Star - My favorite sidescrolling game anywhere ever for reason of its manic pace, enormous diversity of abilities, sublime soundtrack, creative worlds and genre satire, and vibrant and beautiful visuals. (The DS version is a whole lot better, though.)
Super Metroid - My second favorite sidescrolling game, this is another one like KSS that does absolutely everything right, albeit in a different way: Samus doesn't have nearly the moveset of Kirby, but the world itself is designed more effectively to maximize each. It's a darker and moodier game than KSS, though, and it lacks co-op.
Yoshi's Island - Third favorite sidescroller, first favorite pure platformer--as in, not many action elements to speak of. This game screams fun from every orifice, and frankly I'd take it over every 2D and every 3D Mario game combined. One of the best gaming experiences anywhere, though I do prefer the GBA version's controls.
Legend of Zelda: A Link to the Past - Twelve dungeons. What I wouldn't give to have twelve unique dungeons in a 2D Zelda these days. A bit easy, but that's okay since so is every other Zelda past the NES era.
Donkey Kong Country 2 - This is my second favorite platformer, but if I say much more I'm going to start sounding redundant. Again, it just does everything correctly.
Secret of Mana - Now here's a game which has bugs all over the place and a truly broken balance between spells and melee attacks, but I really really don't care because it's awesomely enjoyable even on your own and only gets better still when you factor in three-player co-op.
DoReMi Fantasy - I never knew about this game's existence until Virtual Console, but blang if it didn't swoop in and put an end to my mind's "Super Mario Bros. 3 vs. Super Mario World" debate for third favorite platformer by destroying both of them with its pitch-perfect controls, and magnificently charming and creative worlds.
Super Mario World - Yeah.
I could go on, but let's stop there and name drop a few others without summaries: Earthbound, Final Fantasy V, Illusion of Gaia, Lufia II, Mega Man X 1-3, and Terranigma were some other great, great games.