It comes from the aperiodic structure of real numbers (that is a weird and wrong expression but I cannot find a better way to explain it). Nonrational real numbers are inherently chaotic as seen from the inside (even rational numbers might be seen as such).
Real numbers are weird. Totally.
Also the fact that you are defining something depending on convergence makes it even weirder, as limits do not tend to commute with anything unless smooth conditions take place. And for these fractals, there is no smoothness near the "border".
While "real" numbers are indeed weird, I don't think that's the real explanation here. There are lots of small programs with complex output that have nothing to do with real numbers. If someone gave you one of these programs without telling you what it was for, real numbers would at best be a useful abstraction (bear in mind that there are no actual "real numbers" here, only finite bit-strings). But it might turn out to be the equivalent of a stream cipher, instead. That would, if done correctly, have even more apparent entropy than your typical fractal, which at least has some higher-level structure.
In my view, the real source of complexity is the fact that the algorithm can use arbitrary iterations to magnify small differences. It's the unpredictable nature of computing machines, not real numbers. The nature of real numbers is only a guide to developing and explaining that underlying complexity.
But computing machines aren't unpredictable. They are very predictable. They appear to pull complex structure out of the aether.
Perhaps the interesting thing is that the set of n-bit programs expands to only 2^n possible outputs. Why are some finite number of infinite-length outputs accessible, but not others? Why does nature favor those sequences?
>But computing machines aren't unpredictable. They are very predictable.
Are they though? A 5-state Turing machine might seem trivial, but there exist programs in it, that we don't know if they ever halt or not [1]. Their behavior is totally non-deterministic for our understanding. There are also minimalistic cellular automata that produce completely unpredictable patterns. [2]
There are no non-rational “real” numbers involved whatsoever in drawing this fractal (or e.g. the Mandelbrot set). Only rational numbers, simple purely rational functions, and some amount of truncation of the least significant bits at each step.
The deep “weird” “inherently chaotic” structure you’re talking about is the structure of the rational numbers.
Real numbers are weird. Totally.
Also the fact that you are defining something depending on convergence makes it even weirder, as limits do not tend to commute with anything unless smooth conditions take place. And for these fractals, there is no smoothness near the "border".