What is an example of complex random string, in the Kolmogorov-Chatin sense?
It is impossible to give a specific example of a string with high Kolmogorov complexity. If I was to give an example in this answer, then the following piece of code would retrieve it:

wget http://cs.stackexchange.com/q/9721


(plus some O(1) post-processing).

A string of high Kolmogorov complexity is as elusive as a random number:



There is in fact a connection between randomness and Kolmogorov complexity: a random string has maximal complexity for its length. (At least for one definition of randomness, appropriately called Kolmogorov randomness.)

This observation that it is impossible to exhibit a string of high Kolmogorov complexity can be put into a mathematical form and proved: it is called Chaitin's theorem.

The high-complexity strings are all the ones that you cannot describe.