I've kept most all of my classics over the years. Starting with the SNES, I was too young to give a crap about any of that when I was playing the NES, but even there I have a few.
NES:
Final Fantasy
Dragon Warrior
Dragon Warrior 2
Dragon Warrior 3
Dragon Warrior 4
The Legend of Zelda
Super Mario Bros./Da Cunt
Super Mario Bros. 2
Super Mario Bros. 3
TMNT
TMNT2 The Arcade Game.
Wizardry 4
Metroid
Maniac Mansion
Game Boy:
Final Fantasy Legend 1
Final Fantasy Legend 2
Final Fantasy Legend 3
Super Mario Land
Metroid 2
SNES:
Super Mario World
Super Metroid
The Legend of Zelda: A Link to the Past
TMNT4: Turtles in Time
Final Fantasy 2
Final Fantasy 3
Final Fantasy Mystic Quest
7th Saga
Wizardry 5
Secret of Mana
Dungeon Master
F-Zero
Actraiser
Soulblazer
Mortal Kombat
Mortal Kombat 2
PS1:
Vandal Hearts
Final Fantasy Tactics
Final Fantasy 7
Final Fantasy 8
Final Fantasy 9
Castlevania: Symphony of Night
Dragon Warrior 7
Suikoden 1
Suikoden 2 (yeah that's right, be jealous, that's my crown jewel right there)
Xenogears
Parasite Eve
Alundra
Vagrant Story
Breath of Fire 3
Dead or Alive
Chocobo Racing
Final Fantasy Anthology
Final Fantasy Origins
Final Fantasy Chronicles
Gran Turismo 2
The Legend of Dragoon
Lunar: Silver Star Story Complete
PS2:
FFX
Ephemeral Phantasia
Gran Tourismo 3
Gran Tourismo 4
Suikoden 3
Suikoden 4
Suikoden 5
Xenosaga 1
Xenosaga 2
Dead or Alive 2
Castlevania: Lament of Innocence
Xbox 360:
Mass Effect
Mass Effect 2
GBA:
Super Mario Advance
Super Mario Advance 4
The Legend of Zelda: Four Swords
Donkey Kong Country 3
Metroid Fusion
Metroid Zero Mission
Castlevania Circle of the Moon
Castlevania Double Pack
Fire Emblem
Fire Emblem: Sacred Stones
Final Fantasy Dawn of Souls
Final Fantasy IV
Final Fantasy V
Final Fantasy VI
Final Fantasy Tactics Advance
Golden Sun
NGC:
Super Mario Sunshine
Metroid Prime
Metroid Prime 2: Echoes
The Legend of Zelda: The Windwaker
The Legend of Zelda: Twilight Princess
The Legend of Zelda: Four Swords
Final Fantasy: Crystal Chronicles
Fire Emblem: Path of Radiance
Wii:
Fire Emblem: Radiant Dawn
Super Mario Galaxy
Super Mario Galaxy 2
NEW Super Mario Bros. Wii
Donkey Kong Country Returns
Metroid Prime Trilogy
Metroid: Other M
NDS:
Fire Emblem: Shadow Dragon
Final Fantasy III
Final Fantasy IV
Golden Sun: Dark Dawn
Pokemon Platinum
Pokemon Black
Pokemon Soulsilver
Final Fantasy: Crystal Chronicles: Ring of Fates
Final Fantasy: Crystal Chronicles: Echoes of Time
NEW Super Mario Bros.
Metroid Prime Hunters
The Legend of Zelda: Spirit Tracks
The Legend of Zelda: Phantom Hourglass
Suikoden Tierkreis
PC:
Quest for Glory 1-4
Leisure Suit Larry 1-3
Kings Quest 1-6
Space Quest 1-4
Wolfenstien 3D
Wizardry 7
Eye of the Beholder 1-3
Dungeon Master 1
Dungeon Master 1: Chaos Strikes Back
Dungeon Master 2: The Legend of Skullkeep
Lands of Lore
Doom 1-3
Unreal Tournament
Duke Nukem 3D
Duke Nukem Forever
Quake 2
HeXen
HeXen 2
Heretic
Drakon: Order of the Flame
Warcraft 2: Tides of Darkness
Warcraft 2: Beyond the Dark Portal
Warcraft 3: Reign of Chaos
Warcraft 3: The Frozen Throne
World of Warcraft
Starcraft
Starcraft: Brood Wars
Starcraft 2: Wings of Liberty
Diablo
Diablo 2
Diablo 2: Lord of Destruction
Dungeon Siege 2
Baldur's Gate
Baldur's Gate: Tales of the Sword Coast
Baldur's Gate 2
Carmageddon
Carmageddon 2
Dungeon Keeper
Dungeon Keeper 2
The Sims 1-3
Dragon Age: Origins
Dragon Age: Awakening
Dragon Age 2
... That's all I can think of off the top of my head.
There's probably a few I'm forgetting and there's a lot more I'm not so proud to have and that I don't "cherish". But this is a pretty good list.