TITLE: notation for factoraling a factorial? (since one cannot do n!!)
QUESTION [2 upvotes]: I was thinking about how to get a number to be larger than graham's number very easily... came up with "factoraling" a factorial. However the notation n!! means something completely different. And I don't even think things like n!!! or n!!!! exist... or do they?
Anyway... how would one express or talk about the following?
3!! = 6! = 720
3!!! = 6!! = 720! = broken calculator...
3!!!! = 6!!! = 720!! = broken calculator! = stupidly huge number......
3!!!!...!'s = bazillion times bigger than graham's number
How do I express "n" number of "!'s" after the 3?

REPLY [3 votes]: To take the factorial of a factorial, you should parenthesise, as suggested by arctic tern: $(n!)!$ would be appropriate.
I don't know of any convenient notation for a $k$-times iterated factorial, but that is probably because when dealing with such large numbers, one does not need to be so precise as to differentiate between factorials and exponentials.  In the latter case, Knuth's compact up-arrow notation allows for the expression of very large numbers.  See Wikipedia for details, but essentially:
$a \uparrow b = a^b$ is $a$ multiplied by itself $b$ times.
$a \uparrow \uparrow b = \left. a^{a^{...^a} } \right\} b \text{ times} $ is a tower of $b$ copies of $a$; that is, $a$ up-arrowed by itself $b$ times.
$a \uparrow \uparrow \uparrow b = \underbrace{a \uparrow \uparrow (a \uparrow \uparrow (a ... \uparrow \uparrow a) )}_{b \text{ times}} $, and in general
$a \uparrow^m b = \underbrace{a \uparrow^{m-1} ( a \uparrow^{m-1} (a ... \uparrow^{m-1} a) )}_{b \text{ times}}$.
It boggles the mind (well, my mind) to try to grasp how quickly these numbers grow.
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