KC(x) is always finite for any finite x. What we can say instead is that KC is unbounded. I.e. there is no finite bound on the value of KC(x).

yes, it’s a pigeonhole principle argument. A bit tricky to actually enumerate everything. Imagine KC had a limit k. Then there’s a fixed number of strings that can be compressed to k characters. so considering how there’s an infinite number of strings of length greater than k, they can’t all be compressed to be at most k. Therefore KC has no limit