As Carl said, Kolmogorov complexity is defined mainly for finite strings. It can be extended to other domains of objects but one needs to fix an enumeration of those objects. (An enumeration is a partial function from finite strings to those objects, e.g. Scott mentions two partial enumerations: the language decided by a TM represented by its encoding, and the language recognized by a TM represented by its encoding).

If the domain of objects is uncountable then there is no enumeration that covers all of the objects in the domain so there will be objects left out of the enumeration. For the objects in the enumeration one can define the Kolmogorov complexity of an object as the minimum among the Kolmogorov complexity of its *names*. Those outside the enumeration will have no names (w.r.t. that enumeration) and one can say they have infinite Kolmogorov complexity.

So if you have an infinite domain it will have objects which don't have finite Kolmogorov complexity.

However, the choice of the enumeration is important here and easily change whether an object has a finite Kolmogorov complexity or not. Without fixing an enumeration the Kolmogorov complexity of an object doesn't have a meaning.