Many Googological functions are based on up-arrow notation and related concepts, which stems from using exponents. But wouldn't it be possible to use the same systems with a more powerful base function, even an uncomputably powerful one? For example, you can define a function that is equivalent of up-arrow notation, but for FOOT.

LEG{1}(a, b) = FOOT^b(a)

LEG{n}(a, b) = FOOT^(LEG{n-1}(a, b-1)) (a)

Then, equivalents of functions based in up-arrow notation can be defined using LEG, and so on.

Hyperrecursion of uncomputable functions seems like a pretty powerful thing.