If we’re talking about SERIOUS airplanes, there’s always the Caproni Ca60, the world’s only nonoplane. Sadly it was destroyed on it second flight due to a shifting load. There’s a model of it at the Musée de l’Hydravion at Biscarosse. It also features in the Miyazaki film The Wind Rises (a beautiful film, quite had me in tears), as does Caproni himself.
Another one in the same vein is a German proposal (it was never built) for a transatlantic plane which among other things had a railway inside the wings so the engines could be changed in flight. Unfortunately I’ve never been able to find it again.
I dunno about flying this iitself, but I’d like to try flying from it…
Supermarine Type 559. It’s not fictional, it’s a real design but it was just never built. I’ve had quite a number of conversations about it with an engineer who was on the preliminary design team, circa 1956. He’s still alive and still has lots of ideas. Not sure I’d actually want to fly it but that form of reality makes it interesting to me. I’m not that interested in things that aren’t real in any sense 
If we need to get real I would like to have flown the TSR2 or a fully engined (as designed) Princess flying boat. Both bring back strong memories of my childhood/youth
I remember reading a book in the school library, c. 1972, about the TSR2, and I think it was already cancelled by then.
@ Peter quite right I think it was cancelled in the 1960’s perhaps 67 or 68. I saw it flying at Warton, Lancs and I never understood why it got cancelled after it had been built. Well actually I did it was all about money as usual.
That Supermarine is so cool !
The TSR2 got shut down with all british industry. It makes our (French) eye brows when Bojo starts the new Tempest fighter plane project.
Maybe it will join the fictional types 
It just got bigger and bigger and heavier and heavier. A huge chunk of the fuselage was containing the computer for image processing (for the terrain following radar); today we laugh at that but back then they just could not do it. Apollo was dealing with similar issues and did so successfully but needing much less CPU power. I reckon the Brits could have done the TSR2 10 years later; mid-1970s.
Defence funding is done differently now. Back then it was “cost plus” so the big firms like GEC never had to deliver anything that worked, or worked reliably. They would get paid all the way down the line. Good times It was OK if you never had to fight a war, so wouldn’t work today
Gerry Anderson’s Thunderbird 2
