"Random number" is just a name with a definition, it isn't any more absurd, than asking for square root of -1. In one system of definitions it doesn't make sense (real numbers), in other it does (complex numbers).
You can dismiss question as nonsensical (no, you cannot calculate square root of negative number, there is none), or you can accept new definition that allows you to say something more about a problem, and use it's results, where they are usefull. And Kolmogorov complexity is an usefull definition, maybe not as much as complex numbers, but still.
Actually, read OP's response to my comment (and the following discussion) - my comment doesn't actually invalidate Kolmogorov; it really just frames where Kolmogorov fits within the context of the question. That is, the original question is in fact nonsensical, but Kolmogorov is a useful way of answering the separate-but-very-similar question using the second school of thought that I outlined.
Any definition of 'random' that I am familiar with is only precisely defined when applied to functions, not numbers. Oddly enough, this distinction is not always made clear when outlining the definition, but if you look carefully, you'll see that this is the way that the term is applied. Statisticians are notoriously sloppy when talking about terms, in the same way that computer scientists are comfortable saying that 5x +2 = O(n)... which is nonsense, because you just said that a linear function is equal to a set of functions (TypeError!). 99% of the time, this sloppiness results in no error. That said, you have to remember to be precise with the remaining 1%, because sometimes the loss in precision will lead you to a completely incorrect conclusion.
And you can invent your own definition of random, yes, the same way that you can invent your own number system for numbers like 5/0. But then you have to rebuild the fundamental relationships from scratch (you need to prove that addition works in this new system the way it does for real numbers: 5/0 + 6/0 may not equal 11/0 in this new system, for example).
I'm not that into statistic, but I remember the problems with conflicting views on what "random" means, when I've been writing thesis about PRNGs.
On one hand probability theory says number cannot be random, on the other hand we want to be able to compare randomness of strings from PRNGs to say which is better. Probability theory says PRNGs are not random, 00000000 is no more or less random than 10011010 and that's the end of discussion. But Kolmogorov complexity allows us to at least define, what it means for a string to be random. Probability theory only allows us to compute probability that given string was taken from random distribution, but we already know that PRNGs are not random, so it feels a little artifical to use probability theory there.
That's why Kolmogorov complexity is useful, more so than 5/0 numbers :) Randomness for a string/number wasn't defined before, so there is no conflict, so I don't see why are you insisting that it's nonsensical to speak about random and not random numbers. 2 definitions for 2 different mathemathical objects. Polimorphism :)
But I'm not mathemathician, and I probably forgot many of the things I should know to discuss with you, so I'm open for arguments.
You can dismiss question as nonsensical (no, you cannot calculate square root of negative number, there is none), or you can accept new definition that allows you to say something more about a problem, and use it's results, where they are usefull. And Kolmogorov complexity is an usefull definition, maybe not as much as complex numbers, but still.