The title and subtitle should be self-explanatory. This topic is for recommending games that are, in your opinion, must-play.
I'll begin with my recommendations, by console:
NINTENDO ENTERTAINMENT SYSTEM:
-Final Fantasy III (w/patch)
-STED: Starfield of Memorable Relics (w/patch)
-Pac Man
SEGA MASTER SYSTEM
-Phantasy Star
-Streets of Rage
-Wonderboy III: Monster Lair
-Super Mario Bros. 3
SUPER NINTENDO ENTERTAINMENT SYSTEM
-Super Mario World
-Terranigma
-Seiken Densetsu 3 (w/patch)
-Final Fantasy V (w/patch)
-Fushigi no Dungeon: Fuurai no Shiren (w/patch)
-Final Fight 3
-Super Street Fighter II
-Super Metroid
-Star Ocean (w/patch)
-Tales of Phantasia (w/patch)
-Fire Emblem: Geneaology of Holy War (w/patch)
SEGA GENESIS
-Streets of Rage II
-Shining Force
-Shining Force II
-Phantasy Star II
-Phantasy Star III
-Phantasy Star IV
-Sonic the Hedgehog 3
-Gunstar Heroes
-Dynamite Headdy
-Exile
SEGA SATURN
-Dragon Force
-Panzer Dragoon Saga
-Radiant Silvergun
-Guardian Heroes
-Street Fighter Alpha 2
-Virtual On
NINTENDO 64
-Legend of Zelda: Ocarina of Time
-Legend of Zelda: Majora's Mask
-Super Smash Bros.
-Mischief Makers (GREAT game!)
-WWF No Mercy
-Ogre Battle 64 (JP only, AFAIK)
-Pokemon Stadium
SEGA DREAMCAST
-Shenmue
-Phantasy Star Online Ep. I and II (also for Gamecube)
-Skies of Arcadia
-Power Stone
NINTENDO GAMECUBE
-Ikaruga
-Tales of Symphonia
-Legend of Zelda: Wind Waker
-Baten Kaitos: Eternal Wings and Lost Ocean
-Super Smash Bros. Melee
-Eternal Darkness: Sanity's Requiem
-Fire Emblem: Path of Radiance
-Phantasy Star Online Ep. I and II
SONY PLAYSTATION
Too many to count, but mainly:
-Castlevania: Symphony of the Night
-The King of Fighters '95
-Suikoden
-Suikoden II
-Gran Turismo 2
-Twisted Metal
-Deception II: Kagero
-Threads of Fate
-Front Mission III
-Tales of Destiny
-Tales of Eternia
-Star Ocean: The Second Story
XBOX
-Knights of the Old Republic
-Knights of the Old Republic II
-Otogi: Myth of Demons
SONY PLAYSTATION 2
Again, way too many. In particular:
-Shin Megami Tensei: Nocturne
-Shin Megami Tensei: Digital Devil Saga I and II
-Suikoden III
-Suikoden IV
-Suikoden V
-Suikoden Tactics
-Burnout 3: Takedown
-Jak & Daxter 2
-Dynasty Warriors 5
-Samurai Warriors
-OZ: The Sword of Etheria
-Makai Kingdom
-Gran Turismo 4
-Fatal Frame II/Project Zero II
-Forbidden Siren
-Shadow the Hedgehog
-God of War
-Tales of Legendia
GAME BOY ADVANCE
-Super Robot Taisen Original Generation 1 and 2 (soon in English)
-Yggdra Union (soon in English)
-Tactics Ogre: The Knight of Lodis
-Tales of Phantasia
-Black/Matrix Zero (Japan only)
-Riviera: the Promised Land
-Fire Emblem: Blazing Sword
-Fire Emblem: Sacred Stones
-Fire Emblem: Fuuin no Tsurugi (w/patch)
...there. That's it for now. Feel free to suggest yourselves!
I'll begin with my recommendations, by console:
NINTENDO ENTERTAINMENT SYSTEM:
-Final Fantasy III (w/patch)
-STED: Starfield of Memorable Relics (w/patch)
-Pac Man
SEGA MASTER SYSTEM
-Phantasy Star
-Streets of Rage
-Wonderboy III: Monster Lair
-Super Mario Bros. 3
SUPER NINTENDO ENTERTAINMENT SYSTEM
-Super Mario World
-Terranigma
-Seiken Densetsu 3 (w/patch)
-Final Fantasy V (w/patch)
-Fushigi no Dungeon: Fuurai no Shiren (w/patch)
-Final Fight 3
-Super Street Fighter II
-Super Metroid
-Star Ocean (w/patch)
-Tales of Phantasia (w/patch)
-Fire Emblem: Geneaology of Holy War (w/patch)
SEGA GENESIS
-Streets of Rage II
-Shining Force
-Shining Force II
-Phantasy Star II
-Phantasy Star III
-Phantasy Star IV
-Sonic the Hedgehog 3
-Gunstar Heroes
-Dynamite Headdy
-Exile
SEGA SATURN
-Dragon Force
-Panzer Dragoon Saga
-Radiant Silvergun
-Guardian Heroes
-Street Fighter Alpha 2
-Virtual On
NINTENDO 64
-Legend of Zelda: Ocarina of Time
-Legend of Zelda: Majora's Mask
-Super Smash Bros.
-Mischief Makers (GREAT game!)
-WWF No Mercy
-Ogre Battle 64 (JP only, AFAIK)
-Pokemon Stadium
SEGA DREAMCAST
-Shenmue
-Phantasy Star Online Ep. I and II (also for Gamecube)
-Skies of Arcadia
-Power Stone
NINTENDO GAMECUBE
-Ikaruga
-Tales of Symphonia
-Legend of Zelda: Wind Waker
-Baten Kaitos: Eternal Wings and Lost Ocean
-Super Smash Bros. Melee
-Eternal Darkness: Sanity's Requiem
-Fire Emblem: Path of Radiance
-Phantasy Star Online Ep. I and II
SONY PLAYSTATION
Too many to count, but mainly:
-Castlevania: Symphony of the Night
-The King of Fighters '95
-Suikoden
-Suikoden II
-Gran Turismo 2
-Twisted Metal
-Deception II: Kagero
-Threads of Fate
-Front Mission III
-Tales of Destiny
-Tales of Eternia
-Star Ocean: The Second Story
XBOX
-Knights of the Old Republic
-Knights of the Old Republic II
-Otogi: Myth of Demons
SONY PLAYSTATION 2
Again, way too many. In particular:
-Shin Megami Tensei: Nocturne
-Shin Megami Tensei: Digital Devil Saga I and II
-Suikoden III
-Suikoden IV
-Suikoden V
-Suikoden Tactics
-Burnout 3: Takedown
-Jak & Daxter 2
-Dynasty Warriors 5
-Samurai Warriors
-OZ: The Sword of Etheria
-Makai Kingdom
-Gran Turismo 4
-Fatal Frame II/Project Zero II
-Forbidden Siren
-Shadow the Hedgehog
-God of War
-Tales of Legendia
GAME BOY ADVANCE
-Super Robot Taisen Original Generation 1 and 2 (soon in English)
-Yggdra Union (soon in English)
-Tactics Ogre: The Knight of Lodis
-Tales of Phantasia
-Black/Matrix Zero (Japan only)
-Riviera: the Promised Land
-Fire Emblem: Blazing Sword
-Fire Emblem: Sacred Stones
-Fire Emblem: Fuuin no Tsurugi (w/patch)
...there. That's it for now. Feel free to suggest yourselves!