The Kolmogorov complexity ("K") of a string ("S") specifies the size of the smallest Turing machine that can output that string. If a Turing machine (equivalently, by the Church-Turing thesis, any AI) has size smaller than K, it can rewrite its code as much as it wants to, it won't be able to output S. To be specific, of course it can output S by enumerating all possible strings, but it won't be able to decide on S and output it exclusively among the options available. Now suppose that S is the source code for an intelligence strictly better than all those with complexity <K. Now, we are left with 3 options:
- The space of all maximally intelligent minds has an upper bound on complexity, and we have already reached it.
- The universe contains new information that can be used to build minds of greater complexity, or:
- There are levels of intelligence that are impossible for us to reach.
Keep in mind that adding a 'random number' instruction to a turing machine allows it to create output of infinite complexity, and that pretty much all compute hardware these days contains a hard RNG based on quantum randomness.