The question of the relationship between Kolmogorov complexity and data compression is rather difficult.
However, at the heuristic level, the complexity of an object and the rate of its compression by known algorithms are related.
I used the built-in function Compress of Mathematica to estimate the compression ratio of various data reducing to a character string.
Not for the entire string, but rather gradually, observing how the compression ratio changes with the growth of data volume.
Here's what happened with: Pi digits, some text, Thue-Morse sequence, differences between primes and built-in RNG:
I believe that a significant estimate of data complexity isn't just degree of compression of a whole data, but the parameters of changing it with increasing data length.
But to get these parameters (by fitting eg), we need to know what is the hypothetical shape of the curve?
Power, exponential, linear-fractional or it's combination?
We see that the law is there, but what is the form?