I'll start off with Raiden II's Repeated Tragedy.
A local restaurant had this game, and every time we went, I'd beg my parents for money so that I could play it. I wasn't very good at the game, so I never got past the first stage, and so I heard this song a lot. Really gives me nostalgia.
Also got to shout out Streets, from Timesplitters too. I put a lot of time into this game as a kid, and this song is what got me into futurepop (love me some VNV Nation).
Ah, video game songs that are just plain weird are one of my guilty pleasures. Here's one that lives rent-free in my head.
I found a YouTube link in your comment. Here are links to the same video on alternative frontends that protect your privacy:
I meant to say, holy shit, that was horrific.