The back seat of a police car.
Blue and orange taxis in Tehran.
The Bullet Train from Nagoya to Kyoto.
A “Gull wing” Mercedes
A D’Tomaso Pantera
A lot of helicopters: B-model Huey gunship, d-model huey, AH-1 Cobra gunship, OH-6A, MD500, MD900 NOTAR, SH-3, MI-8, Bell 205, 212, 214.
Military transports, C-130, C-123, C7A
A Jitney, one of those heavily decorated taxicabs that hold about 15 people, in Bali, July 1988.
The Bullet Train (Shinkansen) in Japan, but that’s not too exotic for those who live there. Then again, I suppose nothing is.
An elephant.
I took the hovercraft between Dover and Calais back in 1980.
Do they still have those things? Or did they go the way of the autogyro and the dirigible after the Chunnel was completed?
I wish I could say something like a back of an Emu, but the best I’ve got is a Japanese Acura Vigor. Not all that exotic, except it was a true Japanese model with the steering wheel on the right. And we were driving it in North America. We got some strange looks, which I admit we solicited by doing things like reading the newspaper in the passenger seat.
Ike, I took that hovercraft on that same trip in '83; I’d forgotten all about that. Also a hydrofoil up the Rhine.
A Dolphin! 
Okay, so it was only for about ten yards. It was still a means of transportation, and extremely cool.
Rode on a carriage pulled by a donkey…
A customized 1950 Chevy pickup, a non-miliarty camouflage-painted '49 Jeep pickup, and St. Charles Ave. streetcars
Most exotic for me:
Train through Channel Tunnel.
Horse. (Very exotic for this urban dweller…)
Hydrofoil across Lake Ontario.
Ford Model T
TravelAir biplane
hydrofoil
maglev
I’ve been on an elephant as well! Lots of fun, except that the hair on its neck scratched up the inside of my thighs pretty good (try explaining THAT to your fiancee after a trip to Thailand! “No, it wasn’t a bargirl, honey… it was an ELEPHANT, I tell ya!!”).
That’s about as exotic as my life gets, I’m afraid. 
GusNSpot, riding a steer seems pretty exotic to me. You reminded me that I’ve done a hot air balloon, too. Great ride! I’ve been in a helicopter, too…but under one? Wow!
SPOOFE, a submarine is pretty exotic for most people…unless you mean the old ride at Disneyland.
GingerOfTheNorth, I guess most people would consider a dog sled exotic, unless you live where there are few roads and lots of snow. In these days of endless cities, that’s exotic!
pravnik, did they let ya use the siren? Pretty cool anyway.
Mooney252, riding white water sounds exotic to a fat city boy like me, but sharing a river with grizzlies sounds really exciting! Okay, crazy.
Earl of Sandwich, that wuz just a ride-along, right?
Uncle Bill, tanks for the memory. (Yes, I really said that…I have no shame.)
gatopescado, was that a steam train? Anyway, the Andes location makes it exotic to me.
Davebear, the dolphin really is cool! I thought it was a big deal just to pet one.
Robot Arm, what’s a maglev like? Is it quiet? Where was it?
Sunspace, a horse? The hydrofoil sounds pretty neat, though.
I’m enjoying this. I’m not exactly a shut-in, but it looks like Dopers in general really get around. No surprise there.
Just caught this in Geezer’s reply. I’ve also been in the back seat of a police car. In my case, I was walking home from summer school and stopped on an overpass to rest. This cop came along and gave me a ride home because he was afraid I was going to pass out and tip over backwards.
Anyone in the New Orleans, especially Metairie or Harahan, should be familiar with that overpass; it’s the one that takes Hickory over the railyard. There’s also a cement company at the north end of the overpass and I had slogged through some runoff. I feel sorry for whoever had to clean dried concrete out of that car.
Bamboo raft (across a river in China).
Bamboo fishing boat (watching cormorant fishing at night), in China.
Really gorgous sleeper-car train (loads of handsome woodwork)with coal-fired steam engine, also in China.
Really rickety, ancient, east-german prop jet (yes, in China).
Russian sedan (auto), the old big round ‘bomber’ kind, with the curtains in the windows. China, again. Almost daily for three weeks, with a personal driver. We rated! (guests of the government)
Mongolian pony. (where? you guessed it!)
Hot air balloon. No, not in China. My step-dad was a balloon pilot (licensed, not professional).
I also think I was probably one of the last kids to ride on the back of the galapagos tortoise in the San Diego zoo. I can remember the feel of the shell, sun warmed, and the tip and shift under me as it walked… Back in the days when zoos weren’t quite so sensitive to the animal’s needs…
Segway (well, I didn’t really go anywhere since I ended up where I started :P)
The first flight of a Boeing 777 fully loaded with passengers. Boeing recruited employees for a couple of test flights to check out the creature comforts of the new airplane.
A school bus in a figure 8 race.
How exotic was my old Puch moped?! 
42’ Chris Craft Yacht?
30 foot Cigarette Power Boat?
I’ve been up in hot air balloons many times. But only once in a countach lamborghini – and that was enough for me!
I’ve never ridden an exotic animal. Well, unless you count my Danish husband* – he’s pretty exotic.
*A piggyback ride, people! Get your dirty minds out of the gutter.
The HMS Bounty.
We visited Australia a few years ago and they have the replica of the Bounty that they used in Mel Gibson’s, “Mutiney on the Bounty.” They were doing lunch and dinner cruises up and down Sydey harbor. Pretty neat.