I know the question is rar-EST, but there are just too many one of a kinds, or nearly one of a kinds that I've seen to just narrow it down to one. lol
The only Boeing 337 Stratoliner in flying condition - before she went to the Air and Space Museum. I saw it the very first time while on my very first solo on July 14th, 2000. It was one of the very first test flights of the newly restored aircraft over at Paine Field in Everett Washington. Man was that a great sight to be flying behind in the pattern!!
Saw it a few more times before she headed for the east coast.
Another one was the only Me-262 in flying condition (at the time anyway) also at Paine Field in the Museum of Flight Restoration center.
Today I watched Boeing's modified 747 depart out of Paine Field while I was at work. I literally stopped in my tracks when I saw that beast! I couldn't believe how incredibly huge that thing looked!
At Boeing Field in Seattle the military division of Boeing had a modiefied 757 ( I think it was 757) that had a modified nose. It was a replica of the radome of the F/A-22 Raptor. Later on it also housed the avionics for boings version of the JSF. The aircraft was their "avionics testbed" And it had a crazy canard mounted on top of the fuse. Take a look at the pic here:
http://www.airliners.net/open.file/1115914/M/ They also had another odd 737 parked there that had some huge hump on it. http://www.airliners.net/open.file/0257249/L/
You can see it at the end of the lineup of AWACS.
Also we have an An-2 based here at the field.
The truely rarest aircraft I have seen is a one of a kind. Our FAA Designated Examiner designed and created his own single seat low-wing aircraft. pretty sweet lookin' little guy too! I'll have to get a pic for you guys one of these days!
There is also a resident Travel Air, and several restored Stinson airplanes on field.
What can I say, Seattle is called Jet-city for a reason! And I love it!