> I guess what I'm driving at is that I don't understand how Information Theory is able to account for things like fractals without saying that either the Mandelbrot Set is random or that it contains infinite information. Am I looking at this problem correctly?

This is (I think) the "coastline paradox": coast lines have different lengths depending on scale. Length grows with resolution, so in a sense the length of a coastline can be infinite.

I think the escape hatch is that the information conveyed is not the points on the screen, but the rule on how to generate those points. Going back to my other post this means that the city builder conveys no information about a particular city map.