[uipvjbrn] Gaps in the hyperoperation hierarchy

Iterate f(x)=x+2 and you get a sequence that grows 2n, hand-waving the choice of initial value.

Iterating 2x yields 2^n.

Omitted from the hyperoperation hierarchy is iterating x^2, yielding 2^2^n.

Iterating 2^x yields tetration, a power tower of n 2s, expressed in Conway chained arrow notation as 2 -> n -> 2.

We can make the height of the tower double each time: 2 -> (2^n) -> 2.  (Not a function iteration)

Square the height of the tower each time: 2 -> (2^2^n) -> 2.  (Also not a function iteration.)

Express height of the tower itself as a power tower: 2 -> (2 -> n -> 2) -> 2.  This is a 2-times iteration of tetration.

Beyond here lies madness?

An n-times iteration yields the next hyper operator "pentation" 2 -> n -> 3.

The hyper operators continue mind-boggling larger, 2 -> n -> 3, 2 -> n -> 4,... by iterating the previous one n times.

We can consider tricks like making the third argument grow faster: 2 -> x -> (2^n), jumping up the hierarchy by leaps and bounds.