TITLE: How exactly does Knuth's Up-Arrow notation work?
QUESTION [15 upvotes]: I've done some research, and found this on Wikipedia.
\begin{matrix}a\uparrow b=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix}
\begin{matrix}a\uparrow \uparrow b&={\ ^{b}a}=&\underbrace {a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}} &=&\underbrace {a\uparrow (a\uparrow (\dots \uparrow a))} \\&&b{\mbox{ copies of }}a&&b{\mbox{ copies of }}a\end{matrix}
I'm still a bit confused on this topic. I've done a lot of research, but I can't seem to understand this. Could somebody explain this in an understandable way? From what I can gather, for example 2↑↑3 would be 2^2^2 = 16? Then 2↑↑↑3 would be 2^2^2^2 = 256? Am I correct, or have I mis-interpreted this?

REPLY [16 votes]: Close! The idea behind the up-arrow notation is the so called Hyperoperation Sequence, which goes like:

Successor: add $1$. $S(a)= a+1$
Addition: repeated successor. $b+a = \underset{b \text{ copies}}{\underbrace{ S ( S( \dots S(a)))}}$ 
Multiplication: repeated addition. $b*a = \underset{b \text{ copies}}{\underbrace{ a + (a + (\cdots + a))}}$
Exponentiation: repeated multiplication. $a^b = \underset{b \text{ copies}}{\underbrace{ a * ( a * ( \cdots *a))}}$. This is denoted as $a \uparrow b$, sort of looks like  a^b on a calculator for computing $a^b$. 
Tetration: repeated exponentiation. $a\uparrow \uparrow b ={\ ^{b}a} =\underset{b \text{ copies}}{\underbrace{a^{a^{\dots^{a}}}}} = \underset{b \text{ copies}}{\underbrace{a\uparrow (a \uparrow (\cdots \uparrow a))}}. $ Note that the parentheses are vitally important! $ 2^{2^3} = 2^{(2^3)} = 2^8 = 256 \neq (2^2)^3 = 64$
Pentation: repeated tetration. 

So on and so forth until you get a stack overflow. 
The best way to think of this is as writing a recursive program for a computer. In principle, you could program a computer to compute $2^3$ the following way:
 $$2^3 = 2*(2*2)  = 2+2 + 2+ 2 = 2+1+1+1+1+1+1.$$
As for your example:
 $$\ ^3 2= 2 \uparrow \uparrow 3= \underset{3 \text{ copies}}{\underbrace{2 \uparrow (2 \uparrow 2)}} = 2 \uparrow \underset{2 \text{ copies}}{\underbrace{(2*2)}} = 2 \uparrow 4 = \underset{4 \text{ copies}}{\underbrace{2*2*2*2}} = 16.$$
Notice that we could have been even more recursive in our expansion. 
Moving on to pentation (repeated tetration (repeated exponentiation(...))):
\begin{align*}
 2 \uparrow \uparrow \uparrow 3 &= \underset{3 \text{ copies}}{\underbrace{ 2 \uparrow \uparrow ( 2 \uparrow \uparrow 2) }} \\ 
 &= 2 \uparrow \uparrow ( \underset{2 \text{ copies}}{\underbrace{2^2}}) \\
 &= 2 \uparrow \uparrow 4 \\
 &= \underset{4 \text{ copies}}{\underbrace{2^{2^{2^2}} }} \\
 &= 2^{2^4} \\
 &= 2^{16} \\
 &= 65536.
\end{align*}
Again, notice that we could be even more explicit and break the recursion all the way down to the successor function:
 $$2 \uparrow \uparrow \uparrow 3 = 2 + \underset{\text{many many copies}}{\underbrace{1 +1 \cdots + 1}}$$
Edited to add: more examples!

$n^n = \ ^{2} n = n \uparrow \uparrow 2$ and $n^{n^n} = \ ^{3} n = n \uparrow \uparrow 3$. 
$2 \uparrow \uparrow \uparrow 2 = \underset{2\text{ copies}}{\underbrace{ 2 \uparrow \uparrow 2}} = \underset{2\text{ copies}}{\underbrace{ 2 \uparrow 2 }} = \underset{2\text{ copies}}{\underbrace{ 2*2}} = \underset{2\text{ copies}}{\underbrace{  2+2}} = 4$. Up-arrows to the second is kind of weird. Consider that $a^2 = a*a$ for any $a$. Similarly, $a \uparrow \uparrow 2 = \underset{2\text{ copies}}{\underbrace{a \uparrow a}}$. What can we say, $2$ is weird as an exponent (tetraponent?). 
$3 \uparrow \uparrow \uparrow 3 = \underset{3\text{ copies}}{\underbrace{ 3 \uparrow \uparrow (3 \uparrow \uparrow 3)}} = 3 \uparrow \uparrow (\underset{3\text{ copies}}{\underbrace{3^{3^3} }}) = 3 \uparrow \uparrow 7625597484987$
which is $\underset{7625597484987\text{ copies}}{\underbrace{3^{3^{...^3}}}}$
A googol is pretty big---$10^{100} = 10^{10^2}$. It's so big that writing it as $\underset{100\text{ copies}}{\underbrace{10*10 * \cdots *10}}$ doesn't even really make sense. As big as a googol is, it is absolutely positively miniscule compared to $\ ^{10} 10 = 10 \uparrow \uparrow 10= 10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}$. This is a number that is so big that it doesn't even make sense to write in terms of exponentiation. As gigantic as $10 \uparrow \uparrow 10$ is, it is a fraction of a fraction of a fraction ... of a percentage of $10 \uparrow \uparrow \uparrow 10$, which is so big that expressing it as repeated tetration is impractical and as repeated exponentiation is almost intractable. 
Graham's Number
$1 \uparrow \uparrow \uparrow \dots \uparrow a= 1$. Lame, but consistent: $1^a = \underset{a\text{ copies}}{\underbrace{1*1* \cdots * 1}}= 1$ for any $a>0$. 
$0 \uparrow \uparrow \dots \uparrow a = 0$. Again, $0^a = 0$ for any $a>0$. 
$n!$ grows pretty fast: $1, 2, 6, 24, 120, 720, 5040, 40320, \dots$. But $n! \ll n^n = \ ^{2} n = n \uparrow \uparrow 2.$