To start off, probably the simplest thing that could qualify as a stunt would be to take a ramp and jump over something. And that's always fun, but it's not terribly exciting. What's much more exciting is when you don't actually make it over whatever it is you're jumping. Like this:
There's a lot of great stuff in that video if you feel like watching the whole thing. When you do that many jumps, you find some creative ways to keep it interesting.
In a similar vein, we have the classic "Car meets Fruit Stand" movie cliche, which goes a little something like this:
Moving into the world of video games, here's a cool trick from the wonderfully absurd game ExciteBots: Trick Racing, wherein the lizard robot car uses his giant robotic arm to swing around a huge pole, thereby executing an extremely sharp turn:
This game has a lot of ridiculously over-the-top stuff going on in it. It might not be practical to incorporate giant robotic arms into our game, but there would be plenty of ways to employ a similar mechanic to allow players to swing around sharp turns.
Finally, here's an entry from outside the genre of racing games, Psychonauts' Rolling Tunnel of Crazy Logs Bit (please excuse the congested twelve-year-old who keeps yammering on in the background):
Although this segment is built for third-person platforming, I don't think it would be too difficult to adapt the idea for a racing game; essentially driving along a path laid out in a large tube, you have to maintain a certain speed and stay aligned on the track to keep from falling out.
That's all for now, folks.