There would be way too much for me to cram into a forum post, so I'll hit some highlights. Everything really depends on the mood I'm in. If I'm really engrossed in a particular game series, I tend to listen to the music from that. I've definitely had classic Need For Speed tunes blaring, jazzy tunes from Gran Turismo, and I've even had the orchestral amazingness that is Ace Combat's soundtrack (specifically 2, 4, and 5.)
As for other tunes that aren't compositions made specifically for games:
This one particular song stood out to me the most from GT5's soundtrack because it really felt like the quintessential cruising at night song. I can practically imagine driving a sports car in the darkest hours of the night through the city. Barely any traffic, and just pretty much down to a crawl, just taking in the scenery. On Wednesday nights, I come back into town from the north end from the volunteer thing I have a handful of miles down the interstate because there's a gas station that's pretty well-priced. I fuel my car, and start heading home via main street downtown. I've had that song play as I've driven down main street, and it brings a smile to my face because it recreates what I picture in my mind listening to that song. Sure I'm in a econobox hatchback and not a sports car, but it feels good regardless. Especially when I drive five miles per hour less than the speed limit (don't worry, it's three lanes and there's practically nobody on the road when I do this) and can easily coast through every intersection without coming up on a red light.
Sometimes I'm in the mood for chill hip-hop, other times I like rock and metal, jazz is certainly a favorite.
Then there's a handful of "speed ticket bait" songs. You know, the ones that you can get easily lost in, and when you look at your speedometer, you're going 80mph in a 65 zone. Thankfully I've never gotten a speeding ticket, but lord knows, these songs have made me go "whoa, slow it down" a couple of times:
"Body Burn" - Cubanate
"Ultra" - KMFDM
"N.W.O." - Ministry
"Megalith" - T-Square (I would've posted the original, if it wasn't blocked on YouTube)
https://www.youtube.com/watch?v=Ra6UsC4EM4k
Yeah, that's a small cross-section of my driving music. I could go on for pages about it, there's just so much stuff out there that I like.