Best NES, SNES, N64, and GB/GBC/GBA Games?

Terranigma.

SNES:
Shadowrun: This is in my opinion one of the best CRPGs ever made. I love the setting & atmosphere in this one, and while combat can feel a tad bit clunky, I do feel that the rest of the game more than makes up for it.

NES:
Mr. Gimmick: Might be tricky to get a legal copy of this one, was it was only released in Japan & the Nordic countries. But it's one of the best platformers released for the system. It also sounds great.
Journey to Silius: Another excellent platformer. These days it's mostly known for its soundtrack, but the gameplay is at least as good.
Startropics: I know that you said no Nintendo series, but this one is often overlooked.
Uforia: Another excellent platformer that got a limited release. Not quite as good as the above mentioned ones, but I still quite liked it. Feels a bit like a lighthearted Blaster Master, but without the overheard sections.

Mystic Quest!

Pokemon Trading Card Game for the Game Boy Color?

Super Smash Bros & Super Mario 64 (N64), Supermetroid (SNES).

For the Super Nintendo, I'd recommend the timeless classic, Zombies Ate my Neighbors. Another one I very much enjoy is ActRaiser.

N64: Conker's Bad Fur Day. Had a blast playing it

SNES: Top Gear 2

SNES: Terranigma, Secret of Evermore
Both action-rpg games. i prefer Terranigma.

SNES: Sunset Riders, Super Bomberman 5.

NES: Faxanadu, Contra, Super C, Ninja Gaiden 2, TMNT, Bubble Bobble, Castlevania (1-3), Megaman(1-6), Bionic Commando, Excite bike.

SNES: Final Fight, Ultimate Mortal Kombat 3, Top Gear 2, Super Turrican 2,Super Double Dragon, Blackthorne, Bomberman 2, Sunset Riders, Killer Instinct, Flashback, Rock'n'Roll Racing, Jungle Strike, R-Type3, F-zero, Chrono Trigger, Earthworm Jim 2, TMNT IV, Metal Warriors: Heavy Metal Action, Contra III, Super Castlevania IV, Alien 3, Bust-a-Move, Super Ghouls'n'Ghosts,Zombies Ate My Neighbors.

GBA: xD idk maybe sonic's ones or the wario world, i never played much this platform.

N64: San Francisco Rush 2049 (was the only ive played from this platform aside mario games but is really fun)

Advance Wars

(This is a bit off topic.)

I had a NES, a GameBoy and a SNES back then. And quite a few games. And it's weird; I can't recall any "best games". I can recall a dozen for my Amiga and my first PC, but for the consoles, it's like my memory is blank. Isn't that weird?

Clock Tower and SOS made my poor little Genesis learn what jealousy is.

The original game boy had a great game called "Great Greed". The first RPG I played to death!

On GBA Duke Nukem Advance. My favorite GBA game of all times <3

Tsubasa, Tsubasa 2

Nekketsu Kunio Footbal League 2

TMNT

SNES:
FFVI
Chrono Trigger

GBA:
Yggdra Union
Riviera: The Promised Land - Somewhat strange RPG
Advance Wars (2 is better than 1)
Fire Emblem 6,7,8

Duck Hunt?

GBA-Golden Sun and it's sequel Golden Sun The Lost Age

SNES: Castlevania IV. - Final Fantasy VI
GBA: Advance Wars 2

SNES: mystic quest legends, aka final fantasy 1, NES: kid icarus

I'm going to avoid doubling up on recommendations that are already posted here, but I agree with most of them. Especially the SNES RPGs, those are legitimately some of the best games ever made. Here are a couple more games that weren't listed yet but are absolutely worth playing.

NES:
Adventures of Lolo 3 (There's no running plot or reason to play them in order, so if you like this one go backwards)
Archon
Little Nemo: The Dream Master
Maniac Mansion
Shadowgate (there are two very similar games if you like this one, Deja Vu and Uninvited)
Solstice: The Quest for the Staff of Demnos

SNES:
Illusion of Gaia

N64:
Blast Corps
Bomberman 64
Jet Force Gemini
Mischief Makers
Mystical Ninja starring Goemon
Perfect Dark
Snowboard Kids
Space Station: Silicon Valley

GB:
Final Fantasy Legend 2 (These aren't really Final Fantasy games, they're in the SaGa series)
Final Fantasy Legend 3 (Again, there's no running plot. 2 and 3 tend to be hate one, love the other, so try both)

GBC:
Deja Vu 1+2 (I mentioned #1 up in the NES section, but this is the easiest way to play the sequel)

GBA:
Sword of Mana

Illusion of Gaia. It's actually part of the same loosely related trilogy as Terranigma that others have mentioned. Others have mentioned most of what I would have thought of, but IoG was always a somewhat obscure gem.