That box is merely a proof that the intelligence of patterns in a nonhalting Turing machine needs not be bounded. If we cannot get infinite space/time, we run into sooner problems than Kolgomorov complexity. (As I understood it, OP was about how even infinite ressources cannot escape the complexity our laws of physics dictate.)
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: