I was contemplating the properties of the ultrafuncton (see the previous post), and I was wondering if there might be some higher-level version of the ultrafunction, and I realized that the ultrafunction itself is that function. For example:
u(f(a), n) = f(f(f(f(…a…))) (f applied n times)
But then, it follows naturally that
u(u(f(a), m), n) = u(u(u(…f(a)…), m), m) (ultrafunction applied n times)
So therefore, even though in many nested-function-type situations, there is an infinite hierarchy of such functions (consider Knuth’s up-arrow system), in this case, this single function fills in the hierarchy all by itself.