Let us fix an encoding of Turing-machines and a universal Turing-machine, U, that on input (T,x) outputs whatever T outputs on input x (possibly both running forever). Define the Kolmogorov complexity of x, K(x), as the length of the shortest program, p, such that U(p)=x.
Is there an N such that for all n>N there is an x with K(x)=n?
Remark. If we define universal Turing-machines in a different way, the answer can be negative. For example, consider a U that on input (T,x) simulates T on x if the length of (T,x) is divisible by 100, and otherwise does nothing. One can modify this example in several ways to obtain counterexamples for different definitions of universal Turing-machines.