During a family vacation near the end of my school year, I came up with a convincing extension of the well-known Fast-Growing Hierarchy. I currently call it the Extra-Growth Hierarchy (EGH), but its name is one of about three things that may be revised after discussing with another Googologist. I genuinely hope that this doesn't come off as a naïve invention; I intend for this new fuction to possibly even be in the official growth-hierarchy family (one can dream).

Rules and mathematics

\begin{array}{rcl} \text{Ex}_0(n)&=&2n^2\\ \text{Ex}_{\alpha+1}(n)&=&\text{Ex}_{\alpha}^{\text{Ex}_{\alpha}(n)}(n)\\ \text{Ex}_\alpha(n)&=&\text{Ex}_{\alpha[\text{Ex}_{\alpha[n]}(n)]}(n)\end{array} These first three rules are for single argument function subscripts: the base rule, succession rule, and limit ordinal rule (replace ordinal with the \(\text{Ex}_{\alpha[n]}(n)\)'th entry in its set). Note that when applying the second rule, the function is iterated not n times, like in FGH, but by the previous function-amount of times. The only aspect here susceptible to change is for the base case's expression; I chose \(2n^2\) to resemble f2, or \(2^nn\).

\begin{array}{rcl} \text{Ex}_{\bullet0}(n)&=&\text{Ex}_\bullet(n)\\ \text{Ex}_{\bullet\alpha,\beta+1}(n)&=&\text{Ex}_{\bullet\alpha+\text{Ex}_{\bullet\alpha,\beta}(n),\beta}^{\text{Ex}_{\bullet\alpha,\beta}(n)}(n)\\ \text{Ex}_{\bullet\alpha,\beta}(n)&=&\text{Ex}_{\bullet\beta+\alpha,\beta[\text{Ex}_{\bullet\alpha,\beta[n]}(n)]}(n) \end{array} These latter three are for lists of subscripts, where this extension attempts taking off in growth from FGH: the list's base case, succession case, and ordinal expansion. The bullet denotes the remainder of a list, if existent. The fifth and sixth rules both add extra stuff to the previous entry, prolonging effectiveness. When a limit ordinal is accessed, it is added to the previous entry, perpetuating the ordinals' effects (I call it "ordinal recycling/reshuffling")(β is written first to avoid fixed point weirdness).

Ordinal list example: \(\text{Ex}_{0,\omega2}(1)=\text{Ex}_{\omega2+0,\omega2[\text{Ex}_{0,\omega2[1]}(1)]}(1)=\text{Ex}_{\omega2,\omega+\text{Ex}_{0,\omega+1}(1)}(1)\)

Factoids and ending

•Interestingly enough, Ex_1 can be written as \(2^{2^{2n^2}-1}n^{2^{2n^2}}\).
•Originally, the Extra-Growth Hierarchy had seven rules, which later then progressed to eight. Those eight eventually simplified to six due to redundant logic and similar rules.
•While discovering \(\text{Ex}_1(1)\ll\text{googol}\ll\text{Ex}_1(2)\) was relatively simple, it has also been proven with decimal logarithms that \(\text{Ex}_1(12)\ll\text{googolplex}<\text{Ex}_1(13)\).
As always, discussion over EGH's aspects is greatly appreciated!

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