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See here

http://googology.wikia.com/wiki/Fast-growing_hierarchy

for the definitions of the fast growing hierarchy, chained-arrow-notation and two-dimesnional-array-notation.

The first number is mentioned by Saibian , called "Muffles Monster". If we define $$cg(n)=n\rightarrow n \rightarrow \cdots \rightarrow n\rightarrow n$$ with $n$ $n's$ and $G$ is Graham's number. then Mouffles Moster is the number $M:=cg^G(G)$. I estimated the magnitude of this number as follows : Since $cg(n)\approx f_{\omega^2}(n)$ for large $n$, we have $$M\approx f_{\omega^2}^G(G)=f_{\omega^2+1}(G)<f_{\omega^2+1}(f_{\omega^2+1}(2))=f_{\omega^2+2}(2)$$

For the second number define $f(n)$ to be the value of a two dimensional arrow of $n\times n$ $n's$. Xappol is then defined to be $f(10)$. Now, define $N:=f^X(X)$, where $X=f(10)=$Xappol. So we start with $f(10)=Xappol$ and create an array of $n\times n$ $n's$ , where $n$ is the current number , $f(10)=$Xappol times. Since for large $n$ we have $\large f(n)\approx f_{\omega^{\omega^2}}(n)$ , I estimated $$\large N\approx f_{\omega^{\omega^2}}^X(X)=f_{\omega^{\omega^2}+1}(X)<f_{\omega^{\omega^2}+2}(3)$$

Are these estimates correct , or did I miss something ?

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