That's also the interesting math, so it is worthy of study. But the math that is interesting is the exception.
A "randomly" chosen function from the set of all possible functions is a function with some infinite input that maps it to an infinite output (with any of the infinite ordinals in play you like) where there is no meaning to any of the outputs at all, indistinguishable from random. (The difficulties of putting distributions on infinite things is not relevant here; that's a statement of our limitations, it doesn't make these structures that we can't reach not "exist".)
It's not amazing that if we take a "wrong" turn down the interesting math we end up in increasing levels of chaos. What's impressive is how interesting the not-pure-chaos subset manages to be, and how well it holds together.