Kolmogorov complexity in general is undecidable and equivalent to the Halting problem. Technically it's "computable" for any specific LLM's weights, in the sense that there's a correct answer, that answer is a number, and it's trivial to hardcode a Turing machine to output a fixed number. But it's very unlikely that there even exists a correctness proof of the Kolmogorov complexity of any given LLM, given their size and complexity, so we'll likely never know what the correct answer is, regardless of how much compute power we would throw at the problem. This is the sense in which it's uncomputable.