Honestly?

Mario Kart Wii. Possibly the most fun I've ever had on a split screen.

Whacky wheels in dos.

1.) Richard Burns Rally
2.) Geoff Crammond Grand Prix series (3,3 2000, 4)
3.) Colin McRae Rally 2 (GBA)
4.) Need for Speed: Shift (Android)

Same for me, the sheer amount of fun I had with this game outshines any other racing-type game I've played.

Though Sonic Racing ransformed on Wii U comes pretty darn close, also Mario Kart 8 is looking mind blowing, I can't wait to get stuck into it.

As for non kart based racing, the king of arcade racing for me is good ol' Burnout 2.

I don't like racing games too much (I hate those simulator racing games), but I definitely do like some Mario Kart. Which of them? I'd go with Mario Kart Wii.

One that I think is under-rated is FUEL. I'm not a racing game fan, but I found FUEL to be very interesting/fun. Lots of different vehicles that all handle differently, huge wasteland world. It surprised me how much I liked it. I don't know why it didn't get better reviews than it did.

Outrun 2 (and variations thereof) is my favourite. I also really like the Mario Kart series, Sonic Racing Transformed and Wipeout HD. Yeah, not a fan of realism in my racing games :)

Flatout Ultimate Carnage is my current favorite. I really enjoy the flatout games. Bugbear gets the arcade driving feel and control just right. I would like to try their Ridge Racer sometime based on my experience with the Flatout titles they made. Hopefully I would enjoy it as much as I do those games.

Mario Cart :D