The mass increases as you approach the speed of light, so you never get higher than it. All speeds remain finite.
I had already noticed that and edited my reply.
All others…. I'm going to go ride my bike now. It's been a slice, and I'll play later.
jbazzet: nice drawing, I understand, but I don't understand the part about the wheels being immoble. (I apologize if I'm totally missed your point.)
Chup: I've previsouly defined two method of measuring speed of wheels. I've also given an argument why rotational speed doesn't fit within the constraints of the rest of the question. I can try to explain further, but I think that's the crux of the whole thing.
and, yes - I realize I'll probably never convince anyone. but it is fun.