What is an example of complex random string, in the Kolmogorov-Chatin sense? Any string generated from a PRNG clearly has a very short description.  You need the code for the random number generator, the seed, and then the number of times to iterate.  So, it seems that all such strings have low KC complexity.

If the above it true, then what is an example of a "complex" 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.