This one should be the most familiar to you. The factorial is a function applicable to any non-negative integer n, where it is equal to the number of ways n distinct objects can be arranged. [Source]

For example, n! = n x (n-1) x (n-2) x ... x 4 x 3 x 2 x 1.

First 10 values of n!

1! = 1

2! = 2

3! = 6

4! = 24

5! = 120

6! = 720

7! = 5,040

8! = 40,320

9! = 362,880

10! = 3,628,800

This one isn't as obvious as you might think. n!! isn't n! x (n-1)! x ... x 2! x 1!, but the number of exclamation marks affects the rate n! grows.

For example, n!! = n x (n-2) x (n-4) x (n-6) ...

The number of exclamation marks affects the subtraction of the next n in the product sequence.

Examples:

8!! = 8 x 6 x 4 x 2 = 384

9!! = 9 x 7 x 5 x 3 x 1 = 945

If you were expecting multifactorial to grow faster than normal factorial, well, this is the factorial that does grow a lot faster than normal factorial! This is called a superfactorial, and there are multiple definitions that grow at different paces.

Let's go over the first one defined by Pickover.

The n$ factorial is equal to n!↑↑n! or n!↑↑↑2.

You can imagine just how large this thing can get in only a few entries...

1$ = 1

2$ = 4

3$ ≈ 10↑10↑10↑36,305.3158...

4$ ≈ E1.5218#24 (Hyper-E)

5$ ≈ E2.3976#120 (Hyper-E)

The second definition is by Sloane and Plouffe.

The same symbol, n$, is n! x (n-1)! x (n-2)! x ... x 3! x 2! x 1!.

This one is more controllable but isn't as strong.

1$ = 1

2$ = 2

3$ = 12

4$ = 288

5$ = 34,560

The third definition is by Daniel Corrêa.

Once again, this uses n$ for his definition.

n$ = (11...(n times)...11n) x ((11...(n-1 times)...11n)!) x ((11...(n-2 times)...11n)!^2) ... ((111n)!^(n-3) x ((111n)!^(n-2) x ((111n)!^(n-1)

First few values:

1$ = 1

2$ = 22 x 2! = 22 x 2 = 44

3$ = 333 x 33! = 3!↑2 = 2.081 Tredecillion

4$ = 4444 x 444! x 44!↑2 x 4!↑3 ≈ 9.3704x10↑1,100

5$ = 55555 x 5555! x 555!↑2 x 55!↑3 x 5!↑4 ≈ 3.1549x10↑21,191

There are two definitions of this, and one of them was developed by me!

But here's the first one, which was made by HaydenTheGoogologist2009.

His symbol of choice was n‽.

n‽ = a_n x a_(n-1) x a_(n-2) ... a_3 x a_2 x a_1, where _ is a subscript.

Here are some of the values:

1‽ = 1

2‽ = 4

3‽ = 18

4‽ = 4718592

5‽ ≈ 2.928x10↑183,237

The second definition, which is by me, has some inspiration to Sloane and Plouffe's superfactorial definition.

My symbol of choice: n@.

n@ = n$ x (n-1)$ x (n-2)$ x ... x 3$ x 2$ x 1$.

Here are a few values I calculated with this notation:

1@ = 1

2@ = 2

3@ = 24

4@ = 6,912

5@ = 238,878,720

This is the successor to my megafactorial function.

n}^1 = n! (factorial)

n}^2 = n$ (superfactorial)

n}^3 = n@ (megafactorial)

n}^4 = n# (gigafactorial)

n}^5 = n% (terafactorial)

n}^6 = n? (petafactorial)

This can be expanded into n}^n}^2 and n}^n}^n}^2, which are equal to n}^^2}^2 and n}^^3}^2, respectively.

This can then be extended into n}^^^2}^2, n}{10}2}^2, and even n}}4}2 (which equals n}(n}(n}(n}(4)2})2})2})2}^2)

This was made by me as a result of messing around with Wolfram Alpha.

Using the notation n⍒, it is equal to Product[Product[n!,{n,1,n!}],{n,1,x}] with x being any number.

0⍒= 1

1⍒= 1

2⍒= 2

3⍒= 49,766,400

4⍒= 5.84e250

5⍒= 9.17e10,776

6⍒= 2.00e584,577

In just only a few entries we overflow the standard computation time of Wolfram Alpha with this. Now, sure, how useful is this? I dunno, but it's fun for me to find new ways to create large numbers in only a few entries.

Sources: Googology Wiki, WolframeAlpha.

[THIS IS A SIMPLIFICATION OF WHAT THE FACTORIALS DO.]