So, Fells posted Radical Dreamers, the ending theme to Chrono Cross. Here's an very relaxing instrumental cover. Chrono Cross itself has an astonishingly good score, and I recommend kicking back and enjoying the following:
Scars of Time (opening theme)
Chronopolis
Fort Dragonia
Ceil also mentioned music from Touhou Project. I can state sincerely that nearly every piece of music composed for it is good (the creator, who goes by the pen name ZUN, is best described as a sythesizer virtuoso), but you--yes, you--must hear the following:
Evening Sky
Autumnal Waterfall
Bloom Nobly, Ink-Black Cherry Blossoms
And now for a few miscellaneous pieces of old-school greatness, in no particular order:
Final Fantasy VI - Cyan's Theme
Final Fantasy VI - Shadow's Theme (A ninja with spaghetti-western theme music? Brilliant.)
Illusion of Gaia - The Great Pyramid
Chrono Trigger - The Singing Mountain (A track that never actually made it into the original game, but happily did make it's way onto the DS port)
Donkey Kong Country - Aquatic Ambiance
Zelda II: The Adventures of Link - Palace Theme (If you haven't heard this before specifically, now you know where the music from the Hyrule Temple stage of Super Smash Bros. Melee came from!)
The Legend of Zelda: A Link to the Past - The Dark World (a particularly good orchestral cover)
Xenogears - Awakening
Brave Fencer Musashi - Far Into the Sky (main theme)
Final Fantasy VII - Anxious Heart
Mega Man 2 - Dr. Wily's Castle - Stage 2 (Sing with me now: "Urutora-man! Urutora-man SE-BEN!")
Mega Man X - Intro Stage
Diablo - Tristram
Secret of Mana - Fear of the Heavens (opening/title theme)
Secret of Mana - The Oracle (Some trivia: That rapid-fire "percussion" that pervades the song is based on a form of Indonesian ceremonial chanting known as kecak. It involves a large group of perfomers chanting in sync, and the effect here is decidedly eerie, particularly considering what you fight while this plays. The down-side is because it has such an elaborate sound, there doesn't seem to have been any attempts made to work it into remixes or covers of the tune--despite how faithfully the composer replicated it on that old SNES sound chip!)
Any of those bring back memories?
And since I can't resist, some from newer generations:
Silent Hill 2 - Theme of Laura
Xenosaga Episode I - The Miracle (Latin choirs are not uncommon, it's true. But this is probably the single most epic use of it I've heard--and it's done by the same guy responsible for Xenogears and the Chrono games.)
Xenosaga Episode III - Godsibb
God of War II - Main Theme
Shadow of the Colossus - The Farthest Land