It's not really an exotic, but I did get to drive a white '57 T-Bird that was originally owned by the actress Stephanie Powers. A friend of ours owned it, and gave me a drive one day. I was only 15, but about 2 miles from his house, he let me drive it. It was really cool. Only having a drivers permit, I didn't drive it very fast!
Probably the quickest (and scariest) thing I've ever ridden in was my friends fiberglass Jeep. Around here we have a State off-road sand park (about 5 square miles of beach type sand). His Jeep had an 8500 RPM 302 Chevy (aluminum heads, etc.) and 250 HP Nitrous, 4500 stall converter with a trans-brake. It was 2-wheel drive with 20" wide paddle wheels and no windshield. On a sand flat, he launched it from 4500, and I was holding on for dear life. We only went about 500 feet, but anything that can hit over 100 MPH in 500 feet in sand is impressive. He did take it to an asphalt drag strip once. He was running truck tires, and since it isn't really geared for a 1/4 mile, he hit the rev limiter about 3/4 of the way down the track. He still turned 11.14 @ 117 MPH with it. He was white knuckled when he finished. Needless to say, he only ran it once.