I hate racing games but I fucking love Driver: San Francisco.

Stunts was awesome. Beware the mad skills of Skid Vicious! As I kid a friend of mine and I spent countless hours designing tracks. It was one of those ahead-of-its-time sims, like Forgotten Realms Unlimited Adventures, that had a dedicated community for years to come.

Need For Speed: High Stakes (1999) was probably my favorite. The game allowed you to install it to the hard drive and play it without the game CD...and substitute a music CD in the "car stereo" which would play instead. I remember always playing Soundgarden, Screaming Trees, and Pearl Jam CDs in the CD-ROM drive whilst playing it and now Yield is always (fittingly) associated with this game.

Chocobo Racing.

Moto Racer

No one has mentioned Episode One Pod Racer. It never got really good press (mostly due to the movie) but I've never met anyone who played it that didn't love it. It's actually a really depressing game for me because it doesn't work on my PC and it was a game that I could play over and over and over. Every pod was so different and took totally different strategies to be successful with, not to mention little perks like being shot at by Sand People and the ability to taunt in bizarre alien languages.

Viper Racing
FlatOut 2
Dirt 2

C-a-r-m-a-g-e-d-o-o-o-o-o-o-n

There's racing to be done there? Who knew?! ... Good ole virtual road rage! :)

Grand Prix Circuit. (No, I'm not kidding.) Get off my lawn.

Test Drive Unlimited
than some NFS and Froza

Super Laser Racer! :P

I really liked hi-octane as a kid, but then again wipeout was much better. Also gran turismo.

Flat Out: Ultimate Carnage
PGR 2
used to love Microprose F1 on the Amiga too
Then played a lot of Mario Kart 64 at uni
Iron Man Super Offroad on the C64 deserves a mention too
Also how did they ever get Stunt Car Racer working on the C64?

Does Spy Hunter count?
If not, I was always partial to going the opposite way from everyone in Nascar. Or maybe "racing" away from the cops in GTA.
I guess what I'm trying to say is I never like straight up racing games unless there were other dynamics to them. Nitro, oil slicks, and the like.

Unless an arcade racer is done really well I prefer racing sims so I'd go for Geoff Crammond's Grand Prix 3 / Grand Prix 4 (tough to get to work on modern machines, sadly). The Colin McRae Rally games were also good and I played Formula 1 '97 (on the PS1) to death. I like the sound of TOCA too but I joined GOG after it had been pulled :( GTR Evolution is great too but setting your cars up is incredibly complicated and hard to understand unless you're a racing nut.

Based on very vague memories Gran Turismo (the first one) was great but I thought the initial part of the game in which you had to earn your licences was better than the actual racing part. There were loads of great cars to unlock but in actual races you didn't face many opponents, which I thought was disappointing.

That's one of the very first games I have ever played. I remember there also being a motorcycle version with vaguely similar game mechanics.