Pokemon Red (Game Boy SP + Game Genie) unlocking the unobtainable Mew (which Nintendo only gave away at rare in-person tournament events) at game start:
Zelda Link's Awakening (Game Boy + Game Genie) changing the value of "1" rupee to "255" rupees for effectively infinite currency (the creator isn't aware of the significance of "255"):
Banjo-Kazooie (N64 + Gameshark) replacing the main characters with the Rareware logo and adding an "infinite jump by pressing A" hack, which adds displacement or velocity with a button press (yes, this is totally absurd):
Goldeneye on N64 had some really cool hidden modes that could only be turned on with Game Genie. I don't recall which ones exactly required the Game Genie, but I'm pretty sure there was one that made everyone invisible until their first death in multiplayer. That was a lot of fun.
Some of the hidden stuff in Goldeneye was available without memory hacking, it just required a long sequence of button presses that was unknown for a fair amount of time. I believe stuff like invisibility in multiplayer or the special developer models are only available via button sequences.
I memorized the invisibility sequence. Grabbing Oddjob and smashing in that invis cheat was a sure way to make everyone hate you at a party. Oddjob was a good pairing because you weren't completely invisible, so his short stature/hitbox made the effect more scummy.
You're right that N64 did not have Game Genie (Super Nintendo and Sega Genesis also did), but N64 did have Gameshark, which was the spiritual successor to Game Genie (along with Action Replay). There is a project (https://github.com/JCR64/GoldenEye-007-N64-Gun-Game) that adds games to Goldeneye via Gameshark
Super Mario Bros 3 (NES + Game Genie) equipping Hammer Suit / Invincibility:
https://www.youtube.com/shorts/FUVOM3rxZ5U
Super Mario World (SNES + Game Genie) causing weird memory distortion glitches, enabling swimming anywhere, and other more useful things:
https://youtu.be/o7_eFcpwq24?si=_io5oSzpRxdvuEMp&t=164
Super Mario RPG (SNES + Game Genie) enabling the hidden developer debug menu:
https://youtu.be/bH-uh9BnIfU?si=0L7qviRp5T-Nokjs&t=402
Pokemon Red (Game Boy SP + Game Genie) unlocking the unobtainable Mew (which Nintendo only gave away at rare in-person tournament events) at game start:
https://www.youtube.com/shorts/yzqTRbixnak
Zelda Link's Awakening (Game Boy + Game Genie) changing the value of "1" rupee to "255" rupees for effectively infinite currency (the creator isn't aware of the significance of "255"):
https://www.youtube.com/shorts/dRFqU_Dz3M8
Super Mario 64 (N64 + Gameshark) adding Dorrie the Plesiosaur to Lethal Lava Land:
https://youtu.be/DVBMdwn8Kc0?si=cUZ5ioG1GEvY0l-H&t=84
Super Mario 64 (N64 + Gameshark) showing off the Gameshark menu and a bunch of Mario character animation swaps:
https://www.youtube.com/shorts/AptK2MbqLXQ
Banjo-Kazooie (N64 + Gameshark) replacing the main characters with the Rareware logo and adding an "infinite jump by pressing A" hack, which adds displacement or velocity with a button press (yes, this is totally absurd):
https://www.youtube.com/watch?v=XF6pwm00vtY
Zelda Ocarina of Time (N64 + Gameshark) spawning a hidden developer-only enemy, the Starfox Arwing, which shoots lasers at Link:
https://www.youtube.com/shorts/KOQUhm1cNu8
Smash Bros 64 (N64 Emulator) spawning Master Hand (final boss) as a playable character in a standard stage:
https://www.youtube.com/watch?v=A1i9TRLifaw
Zelda Majora's Mask (N64 + Gameshark) spawning special items, equipping the wrong items, using the wrong weapons, manually adjusting the clock, etc.:
https://youtu.be/ae2q9CjqXsc?si=IPB7XcPFWZyzpbh3
Smash Bros Melee (Gamecube emulator) spawning infinite items and damage for total chaos:
https://www.youtube.com/watch?v=IDZZiL4rqlk
Quick Game Genie hardware overview:
https://www.youtube.com/shorts/WzHokvjdvDc