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The Kolmogorov complexity of an object is measured as the length of the shortest program that is able to output a description of the object. Although of great theoretical importance, I think it has never quite attained as much practical use, mainly for two reasons. First, apart from simple schoolbook examples, it may be extremely hard or impossible to find the shortest conceivable program that produces the right output. If it were easy we would have an optimal compression algorithm. Second, Kolmogorov complexity doesn't always correspond very well to how we perceive complexity. Take for example a program that runs a dynamic system which may bifurcate between a periodic and a chaotic solution. Clearly the chaotic orbit has a much more complex appearance than the periodic orbit, yet the code length is identical in the two cases.
Musical complexity is even trickier to define. It has to take the multiple domains of rhythm, harmony, melody, timbre, articulation, and texture into account. Furthermore, the complexity may play out at a certain temporal scale. For example, in a techno track the timbral and rhythmic structure within a single measure might be quite complex, but the redundancy caused by repetition would make it simple over any longer time-span.
The algorithmic composition Kolmogorov Variations pits these two complexity measures against each other. The goal is to make complex music with programs of short code length. Whether or not the attempt is successful is open for debate, but in any case it is clear that all pieces were made from scratch, just by programming every aspect of them. (Well, apart from some post-processing.) And the code is available for download:
Two journal papers dwell longer on the topics of dynamic systems with multiple time-scales and the feasibility of having simple algorithms creating complex music, they can be found in the above link. And from today on the album Kolmogorov Variations is apparently available on all those popular streaming services.