A Harrier Jump Jet. Though are there even two-seater Harriers?
Though this does look like fun 
https ://www.youtube.com/watch?v=0kDKycNbopM (link broken after https)
its an Evo rally car, if you don’t want to click on a link
Do I have to go to the nearest airport or can I, as a Washingtonian, request to be dropped off at Dulles, near the other Washington?
Since some others have chosen airbourne methods of transport, I’ve always wanted to fly in a DC3.
For land (and water) based locomotion, how about an Amphicar? We lived on a lake in the early 60’s and one of the other lake dwellers was a car salesman. We’d see him out there occasionally, giving potential customers a test drive.
Yeah, they’re not fancy or speedy but it would be fun watching the faces of people who never knew these things existed. And we could putter up Puget Sound (on a calm day) on the way to Sea-Tac (if we have to go to the nearest airport.) Though Boeing Field isn’t much farther and we could cruise on up the Duwamish River, too.
Adam West’s Batmobile.
Just give me a comfy seat with a friendly driver please.
Aston Martin DB5
Any vehicle currently in existence? I was going to say a Harrier II jumpjet, but I think a F-35B. Now THAT would be a hell of a lot of fun. Even better if we can just do a small, tiny, itsy-bitsy strafing run of the congested traffic. Just a small one…
(It’s only the one time, after all)
Finally, someone found the loophole I deliberately placed in the OP. No, it doesn’t have to be the nearest airport. 
Let’s go the full Monty,
I want to be personally chauffeured by the Captain of a Nimitz class aircraft carrier.
Take the damm airport with us to the Caribbean for a joy flight at Mach 2 or summat.
Aren’t those single-seat?
I’m totally not interested in cars, so I just want comfort. But I’m also not interested in contrived conversation. So, any car and I’ll have earbuds in place.
Space Shuttle. If it can’t be launched, I’ll take the ground route.
A late 60s VW Beetle.
It says a ride to the airport, not that you’re taking a flight. Maybe you just like to people watch, or you’re dropping off a delivery.
Yeah, they are. I was hoping they had a trainer variant, but doesn’t look like it. Perhaps they could fly me remotely. 
Well, there goes that dream. I’d settle for the Harrier II (with full load out for that strafing run).
If I can’t have that, then, as the OP seems to be saying we don’t actually have to go to the air port, I’ll take a ride on the first manned SpaceX flight to the ISS as my Uber ticket.
Yeah. Swing by and pick me up!
Y’all got Beck’s scenario all wrong, her driver will get pulled over and arrested by her friendly state policeman and she’ll have to drive herself the rest of the way.
Me? I’m mooching a ride on the aircraft carrier
For the ride to Ohare airport, I’d definitely pick whatever car is used in Ferris Buellers Day Off.
I’m going to have to say a Maybach Exelero.
A Falcon Heavy, taking the scenic route via the moon.