These are by no means an all-time 10 greatest for me, just some I've really dug over the years. Had descriptions for some of these ready, but they just got sillier as they went along, so I've omitted them entirely.

We Are Finally Cowboys - No More Heroes
http://www.youtube.com/watch?v=FISXJNCcLM0


Bloody Tears - Super Castlevania 4
http://www.youtube.com/watch?v=yoQex8CB9eA



Geese Ni Chusshite - Fatal Fury 3
http://www.youtube.com/watch?v=Uchoj7Z2Cwo



Prologue (at least I THINK that's the official name) - Ninja Gaiden 2
http://www.youtube.com/watch?v=TmS80HRjrks&feature=related



Intro (S.O.R. Super Mix) - Streets of Rage 2
http://www.youtube.com/watch?v=DxvIyRcnMbU



In the Earthen Womb - Illusion of Gaia
http://www.youtube.com/watch?v=KelnHjOateU


Mighty Obstacle - Ys: The Ark of Napishtim
http://www.youtube.com/watch?v=9DfzxdyfRO4&feature=related


Vortex Infinitum - Guilty Gear XX Reload
http://www.youtube.com/watch?v=Xr-182Rg9r8


Thermosphere - Einhander
http://www.youtube.com/watch?v=7hJbe7YXsLU

Goodbye Esaka - The King of Fighters 2000
http://www.youtube.com/watch?v=FgI3q3LL5u8&feature=related (arranged)
http://www.youtube.com/watch?v=LLTf0xGW4BE&feature=related (original)

Honorable mentions (stuff that I feel I'm biased towards due to playing the games in question again recently...hell, one of them JUST came out)

Somnia Memorias - Parasite Eve
http://www.youtube.com/watch?v=83-1RU1V5Nk&feature=related

Kline's Dream - Alundra
http://www.youtube.com/watch?v=B3xKuSAqZ-4

Untamed Heart - Klonoa
http://www.youtube.com/watch?v=2NGWhKhMojQ

Cheap Shop - Scott Pilgrim Vs. The World: The Game
http://www.youtube.com/watch?v=uDY5eJa9NxE

Sorry that was so long. =<