Any natural number can be regarded as a bit sequence, so inputting a natural number is the same as inputting a 0-1 sequence, so NP-complete problems with natural inputs obviously exist. But are there any natural problems, i.e. ones that do not use some encoding and special interpretation of the digits? For example "Is n a prime?" is such a natural problem, but this one is in P. Or "Who wins the Nim game with heaps of size 3, 5, n, n?" is another problem that I consider natural, but we also know this to be in P. I am also interested in other complexity classes instead of NP.
Update: As pointed out by Emil Jeřábek, given $a,b,c\in \mathbb N,$ to determine whether $ax^2+by-c=0$ has a solution over the naturals is NP-complete. This is exactly what I had in mind as natural, except that here the input is three numbers instead of just one.
Update 2: And after more than four years waiting, Dan Brumleve has given a "better" solution - note that it's still not complete because of the randomized reduction.