Almost 30 years ago, I read a book that showed you could create an astronomically (actually that word understates it) large number in eight characters: 9!^9!^9!
To get a sense of how big it is, consider nine factorial equals (9*8*7*6*5*4*3*2) = 362,880, so it’s 362,880 ^ 362,880 ^ 362,880.
That means 362,880*362,880*362,880…*362,880, where the first 362,880 is term 1, and the last is term 362,880. And whatever that absurdly large number is, multiply it by itself 362,880 times.
What does that look like? Let’s simplify and call it 300K instead of 362,880, though the compounding effect of the extra 62.8K would be massive. Even 300K^2 (300K squared) is 90 billion. And 300K ^3 = 27,000,000,000,000,000 or 27 quadrillion. And we’re taking it to the 362,880th power. And that’s before we raise whatever absurdly large number we get itself to the 362,880th power.
To simplify more, we know that a number raised to a power equals two of its factors raised to the same power and multiplied by each other. For example, 12^2 = 144, and 3 *4 = 12, so (3^2) * (4^2) also equals 144. So 300K^300K (again simplifying) = 100K^300K * 3^300K. Unfortunately, even the 3^300,000 term is so huge, it’s virtually uncomputable. To illustrate, 3^100th power is 5.15 x 10^47th, i.e. 515 with 45 zeroes after it. But that’s only the 100th power, not the *three hundred thousandth* power. And that’s the tiny term. The 100,000^300,000 term, which is a 1 with 5^300,000 zeroes after it, is vastly bigger — so much so that the number of zeroes in it is far greater than the entire first number.
But this only describes the first two terms, i.e, 9!^9! Once we get this gigantic number that I can’t even compute, we then have to multiply it by itself 362,880 times which is so uncomputable, it’s difficult to wrap one’s mind anything but the process of creating it.
Still let’s take this further. What if we labeled the first 9! “term 1”, the second “term 2” and the third “term 3.” And then we added another term. So it would be 9!^9!^9!^9! Now we’re taking the insanely large number we couldn’t compute from before and multiplying that by itself 362,880 times. But why stop at four?
Instead we could write it as 9!^9!^9!^…^9! where the last term is the 9!th one in the series. In other words, instead of having three or four terms, we have 9! terms. Now we’re cooking with gas. Well, sort of.
Let’s say whatever insanely absurd number generated by this series of 9! terms = x. We could then substitute x which is so incalculably large, bigger than the possible permutations of atoms in the universe by a mile, into the initial equation. So it would be x!^x!^x!^…^x!. Instead or starting with nine, we start with x, and we do it until we get to the x!th term in the series.
If we were to somehow compute the result of this series, we would get some preposterous number. Let’s call that number y. Then we take y and sub it back in for x, and repeat the series to the y!th term, compute it, call it z.
Okay, now let’s call 9 the first term, x the second term, y the third term… and do this whole process z times. In other words, not only are we starting with bigger and bigger initial factorials, we’re doing the substitution process that many times. Call the result of all of that q.
Maybe a picture will make it clearer:
But we’re not satisfied with q. It’s too small. To get a bigger number, let’s create a new operation. Instead of factorial which is symbolized by an exclamation point, let’s use a double exclamation point called q-factorial. And while x! means (x*(x-1)*(x-2)*…*(x-x+2)), x!! means (x*(x-1/q)*(x-2/q)*(x-3/q)*…*(x-q/q)*…*(x-(x+1/q))). In other words, instead of subtracting one from each successive integer and multiplying it as you would with normal factorials, you’re subtracing the tiniest infinitesimal fraction from each successive integer and multiplying the whole way down. If q were 2, for example, it would be (9*8.5*8*7.5*7…), but q isn’t 2, it’s some absurdly enormous, unsayable number, so the number of multiplications is mind breaking.
So instead of 9!^9!^9!, let’s restart the whole process with 9!!^9!!^9!! and run it again. Let’s assume the unsayable-even-by-God number is n. Let’s add a new operation called n factorial, or three exclamation points that subs in n for q. And start over again.
Let’s assume the initial operation with one exclamation point was term 1, q factorial was term 2, n factorial term three, and do this substituting of the factorials n!!! times. Let’s call that number k.
We could obviously take this further, but by now my brain has been reduced back to the primordial soup from which it evolved, and I don’t have more than the vaguest concept of even the process, so we’ll stop.
I’ll just add that if we multiply whatever this monstrosity k is by zero, it’s still zero. Or if we were to say that if a bird scraped its feathers against a giant mountain once every k years, given an infinite amount of time, it would wear the mountain down to dust.