I invented with some extensions of factorial:

n!! = (...(n!)!)...)! (n !'s)

n!!! = (...(n!!)!!)...)!! (n !!'s)

n!m = n!!!...!!! (m !'s)

n!m = (...(n!m-1)!m-1)...)!m-1

And function:

f(n) = n!n!n!...n!n!n! (with n (n!)'s)

Firstly compute n!, this is a number of !'s in the next number, compute it, then this is a number of !'s in the next number ...

f(1) = 1! = 1
f(2) = 2!2! = 2!! = (2!)! = 2
f(3) = 3!3!3! = 3!3!6 = 3!3!!!!!!

Already 3!!!!!! might be very large, and so f(3) can't be easily expressed using conventional notations. To compare it to up-arrows, observe that:

n! < n^n (n>1)
n!! < n[3]n, using Steinhaus-Moser notation.
n[3]n < n^^(n+2)
n!!! < n[4]n
n!!!! < n[5]n
n!m < n[m+1]n < n^m(n+2)
3!!!!!! < 3^65
= 3{6}5

f(3) < 3{3{6}5}5

This is rather good upper bound, I can say that f(n) ~ {n,n,1,2}, but f(n) is "slightly" larger, since factorial grows faster than exponent, and n!! grows faster than tetronent.

The next I find that f(n) > n{n{n...n{n!}n...n}n}n (n n's from the centre out, topped off with factorial of n!)

Finally, I compare it to fast-growing hierarchy: f(n) ~ f_w+1(n).

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