User blog:When i think about you i touch my elf/The Fractal-Growing Hierarchy (FGH2)

Base Rules:
Note: I am using the Greek letter psi because it branches out, just like I intend the FGH2 to branch out, owing to it's fractal nature.
 * \(\Psi_0(x,y) = x^y\)

Recursive Rules:
Note: This isn't an inherently strong hierarchy, but mostly just a way to get a first foot in the door when it comes to making notations.
 * \(\Psi_1(x,y) = \Psi_0(\Psi(x,y),\Psi(x,y))\) for \(p=1\)
 * \(\Psi_{p}(x,y) = \Psi_{p-1}(\Psi_{p-2}(x,y),\Psi_{p-2}(x,y))\) for all \(p>1\)

Ordinal Rules:

 * \(\Psi_{\alpha}(x,y) = \Psi_{\alpha -1}(\Psi_{\alpha -1}(x,y),\Psi_{\alpha -1}(x,y)) if \(\alpha\) is a successor ordinal directly following a limit ordinal. (\(\alpha -1\) equals a limit ordinal.)
 * \(\Psi_{\alpha}(x,y) = \Psi_{\alpha -1}(\Psi{\alpha -2}(x,y),\Psi{\alpha -2}(x,y))\) if \(\alpha\) is a successor ordinal such that \(\alpha -1\) is also a successor ordinal (not a limit ordinal).
 * \(\Psi_{\beta}(x,y) = \Psi^{x}_{\beta [y]}(x,y)\) where \(\beta\) is a limit ordinal and \(\beta [n]\) refers to the nth element in the fundamental sequence of \(\beta\). (Which of course hasn't been defined, but it's assumed to be the Wainer hierarchy, or really whatever else you'd like.)