The original Spyhunter
Final Fantasy games (any really, nearly all make me hum)
Zelda (my mobile phone ringtone... hehe i am so sad )
Super Mario World
Streetfighter II (esp Ryu stage!)
For proper music, it has to be Jet Set Radio Future.
As for regular old "videogame music", here's a few greats:
Punch-out (Arcade and NES)
Legend of Zelda (NES)
Zelda II (NES) (this music is awfully hard to get out of your head; be warned!)
Moon Patrol (Arcade)
Mega Man (most of the series had great music)
Also, if you haven't heard of them, the Minibosses do some great updated NES music. (this is a must-hear band for retro gamers)
There was an early scrolling shooter on the Amiga & ST that I unfortunately cannot remember the name of
More recently, I would suggest the Project Gotham Racing and Gran Turismo series. And surely you couldn't leave out the Grand Theft Auto games? [edit: okay, it's not original, but it's still good music in a game!]
Also, it wasn't a game, but if anyone remembers SAE's Amazing Tunes 2 demo on the Amiga? Smashin'
Oh, I've just remembered... that little game/level opening tune in Galaxians. Short, but very memorable.
Off Topic -- Talking of 'short', didn't the bloke who composed the original Channel 4 theme tune end up earning loads from the royalties every time it was played, god knows how many times over the years. And it was only FOUR notes!
Agreed on SF2 Ryu + Guile. I actually once had a CD of the complete SF2 Arcade soundtracks properly done. It was a freebie with Nintendo Magazine System in about 1992/93. Don't know what happened to it though.
Xenon II was class.
Chrono Trigger for the SNES was very good (especially the "medieval" era map screen theme).
The music for the second world in Legend of the Mystical Ninja used to get stuck in my head, and wouldn't leave for weeks!!
I also remember the music for the hang-glider sections of SNES Pilotwings being absolutely bang-on for the mood of it - really airy and relaxed. Aaah, happy memories. I feel a Snes9x session coming on!!!
1 Secret of Mana - Snes
2 Monty on the Run - C64
3=Dynamite Dan/Manic Miner - Spectrum
All simply stunning and the last two are remarkable for the limited space available in the 8bit era.
Secret of Mana is simply superb in every way, anyone who hasn't heard the music in these games get an emulator and stick them on. Or better still go out and buy them, nothing in this or last generation comes close in musical terms IMHO