I was studying Knuth's up-arrow notation and I was wondering if ever something like "Knuth's up-arrow factorial notation" has ever been used. Now I know this probably isn't a recognizable term, because I made it up. My question is basically asking if something like that(possibly with a different name) has ever been used to create enormous numbers. I'll explain how it works:
Knuth's up-arrow notation works like this:
You have a↑b which results to ab or a * a * a ..... * a where the amount of a's are equal to b.
For example: 4↑3 would be 43 or 4 * 4 * 4 or 64.
Using double arrows is where the numbers start getting big. a↑↑b is equal to aaa..a where the amount of a's is equal to b.
For example: 4↑↑3 would be equal to 444 or 4256. That's a big number!
Enough explaining that. What I like to call "Knuth's up-arrow factorial notation" is where we use arrow notation but b is changed to b factorial.
For example: You have a↑!b which results to ab! or a * a * a ..... * a where the amount of a's are equal to b!.
So, 4↑!3 would be 43! or 46 or 4096. Double arrows get even crazier. 4↑↑!3 would be equal to 444444 which is an enormous number! To simplify that without using scientific notation would be 4444294967296. Basically it's just like Knuth's up-arrow notation but with the b variable put to a factorial.
So what I'm asking is if something like that has ever been used before or if it's something that could be put to practical use.
AFAIK, such "factorials" are not named, and as far as constructing large numbers go, they are quite small. It appears as though all you've done is
$$a\uparrow^k!b=a\uparrow^k(b!)$$
which is a seriously pointless thing. I doubt on an practical use, even for the sake of making large numbers, since
$$a\uparrow^{k+1}b\gg a\uparrow^k(b!)$$
That is, your factorials don't add anything onto the strength of Knuth's up-arrow notation.
And on top of that, what's the point of making a new notation when we can already write it easily?