top 10 objects that don't exist
Well, there's this thing called Locus Theorem, which states that if any three of you can construct a proof that a fourth cannot, then the fourth one doesn't exist!
If you make the first three people, A, B, and C, and say "A cannot make D and E," B can say "I can make D, so D doesn't exist"; C can say the same thing for E, so E doesn't exist, and since A and C know E doesn't exist, E itself doesn't exist.
In particular, it seems to rule out anything that has a non-trivial amount of structure; it can't be "made out of smaller bits of stuff that each have smaller structures, and those have smaller structures, and so on," because it's impossible to build it that way.
As for my top 10 list, here goes --
Top 10 in no particular order:
Matter
Time
Space
Chaos
Quantum particles (i.e. quarks, photons, gluons, etc.)
Phenomena (in the general sense that we're talking about anything that exists)
People
The Big Yud Kavli Prize
Yudkowsky.
