Approximating the Kolmogorov complexity
I guess one possible answer to your question is this: Take a pseudorandom number generator $G$. Try to chose a generator which has some powerful attacks against it: a random number generator attack for $G$ is (for our purposes), an algorithm $A$ which, when given an imput string $s$, determines a seed $A(s)$, such that $G(A(s))=s$. Then approximate the KC of $s$:

input: s
Compute A(s);
if |A(s)| + |G| &gt; |s| output: |s|
otherwise output: |A(s)| + |G|


Where $|G|$ is the length of the program that computes $G(s)$ (often quite short, as for linear generators). 

Note that in practice, random number generator attacks are not as described: they may fail or produce incomplete results. In that case, you may adapt the algorithm so that it returns $|s|$ when the result of the attack is unsatisfactory. The same remark goes for compression algorithms. 

The caveat to this approach as opposed to compression algorithms is that compression algorithms are in general much more suited to computing KC as they are tailored to work on any string, whereas an attack can work only if $s$ happens to be in the image of $G$ (very unlikely).