Handwritten example of developed "faxponential" notation

In reconsidering the "hyperfactorial" operation sets I developed a year or so ago

(this was in the shower, as usual, where I do most of my mathematics, writing and musical composing)

(I must develop a water-proof workstation someday)

I developed a fun little operation that seems to grow faster than factorial series, or even hyper factorial series. Indeed, the operation generates ridiculously large numbers so quickly I have hope that it can economically surpass even values like Eddington's 10^80th power with ease.

Consider the "faxponential" operation, N^!

Now, the "!" symbol usually represents a factorial operation, e.g., 4! would = 4x3x2x1 or 24. 5! = 5x4! = 120, and so on.

Factorial operations grow larger than even exponential functions; they are very unstable beasties.

In contrast, the "faxponential" operation makes a routine factorial series look positively serene.

The "^" operation is the lazy computer programmer's (i.e., my) way of indicating a power operation. E.g., 3^4 = 3x3x3x3 = 81.

Finally, my innovation, using examples, because the notation is very clumsy to write on a word processor.

1^! = 1, by definition. This is the same as the definition for "1!".
2^! = 2^1 = 2
3^! = 3^(2^1) = 3^2 = 9

I think you can see where this is going.

4^! = 4^(3^(2^1)) = 4^9 = 262, 144

This suggests that 5^! is monstrously large, = 5^(262,144)

And even the nice, modestly denoted 10^! is, perhaps, inconceivably large.

I call them "Star Wars" numbers, after their resemblance to the traditional opening scrolling prologue in the Star Wars movies.

Hey; I can do that if I want. They're my numbers!

***

This is an example of mathematical "poetry" as opposed to "prose." Where math is practicable, I think of it as a nice technical manual. But where it is just uselessly cool, that is poetry. Oddly enough, the poetry of math sometimes ends up being practicable somewhere down the line, to some other guy in some other branch of science.

So — maybe — I have just advanced some field, somehow. Cryptography, perhaps? Computer science?