Within the context of the game:
NiGHTS - Dreams Dreams, also the Christmas NiGHTS remix
Parappa the Rapper - All the Master's songs
Sonic CD - Believe
Daytona - Sky High
Burning Rangers - Angels With Burning Hearts, We Are Burning Rangers (both Japanese and English versions)
King of Fighters 94/96 - Psycho Soldier
Fighters Megamix - Rent A Hero
Sonic Adventure - I Wanna Fly (Tail's theme), Amy's Theme, Open Your Heart, Knuckle's theme
Sonic Adventure - Themes for Shadow and Knuckles, all of Knuckle's stage (bad, yet so good)
Sonic R - All stage music
Steep Slope Sliders - Hold Me Close, Memories (?)
Outside of the context (stuff i might listen to if i had never played the game):
Burning Rangers - I Just Smile
Panzer Dragoon Saga - Sona Mi Areru Ec Sancitu
Xenogears - Star of Tears
Zone of Enders 2 - Flowing Destiny
Sega Touring Car Championship - Don't Drop Me
Breath of Fire 3 - Pure Again (never played the game, but it's beautiful)
Parasite Eve - Somnia Memorias (and also the remix of this song on the PE Remixes disc)
Zap! Snowboarding Trix '98 - Practice mode BGM
Steep Slope Sliders - Hold Me Close, Memories (?)
Jet Set Radio - Magical Girl, Super Brother (Guitar Vader has some other great songs between their otherwise hit-or-miss CDs)
Jet Grind Radio Future - Baby-T, I Love Love You, Rockin' the Mic, Butterfly, Concept Of Love
Dance Dance Revolution - Stomp To My Beat, Get Up 'N Move
i'm sure there's tons more, and the second list should be included in the first, but truthfully, a lot of songs would lose their value had i never played the games they were in.
Edit: Someone requested the MGS3 "Snake Eater" song ealier:
Abstracted Camouflage - Snake Eater