Earlier quoted context omitted.
Look at primitive recursive functions. E.g. see http://en.wikipedia.org/wiki/Primitive_recursive_function
Even if we limit it to primitive recursive functions, how can we put a bound on the Kolmogorov complexity of one? We'd need some way of coming up with bounds on the smallest Turing machine to compute that function.
That might be possible. Primitive recursion is much less powerful than Turing machines.