Applications of general arrow notation?

I don't know where to put this, so I put it here cause it is so advanced...

Anyway, for those who knows what arrow notation is, I have some questions, and for those who don't know what it is, here's a brief explanaition:

The arrow notation $\displaystyle a\uparrow b$ is used to apply an operation multiple times ($\displaystyle b-1$ times) at variable $\displaystyle a$ with variable $\displaystyle a$. For example, $\displaystyle a\uparrow^n b$ means that operation $\displaystyle \uparrow^{n-1}$ is applied $\displaystyle b-1$ times at $\displaystyle a$, with $\displaystyle a$ itself. So since $\displaystyle \uparrow^0$ is defined as multiplication, $\displaystyle \uparrow^1$ means power since $\displaystyle a$ is multiplicated $\displaystyle b-1$ times with itself.

$\displaystyle a\uparrow^n b\ =\ \underbrace{a\uparrow^{n-1} a\uparrow^{n-1} a\uparrow^{n-1} ... \uparrow^{n-1} a}_{\uparrow^{n-1} \text{ comes } b-1 \text{ times}}$

$\displaystyle a\uparrow^1 b\ =\ \underbrace{a\uparrow^0 a\uparrow^0 a\uparrow^0 ... \uparrow^0 a}_{a \text{ comes } b \text{ times}}\ =\ a\cdot a\cdot a\cdot ... \cdot a\ =\ a^b$

All this also means that $\displaystyle a\uparrow^n b\ =\ a\uparrow^{n-1}(a\uparrow^n(b-1))$. For example, $\displaystyle a^b\ =\ a\uparrow^1b\ =\ a\uparrow^0(a\uparrow^1(b-1))\ =\ a\cdot a^{b-1}$.

This can go on forever, meaning that $\displaystyle n$ can grov from 1 to 2, from 2 to 3 and so on, creating new operators $\displaystyle \uparrow^n$ all the time. For example

$\displaystyle \begin{array}{ccccl}a\uparrow^0b&=&&=&a\cdot b\\

a\uparrow^1b&=&a\uparrow b&=&a^b\\

a\uparrow^2b&=&a\uparrow\uparrow b&=&\overbrace{a^{a^{a^{a^{a...}}}}} ^{\begin{array}{c}_{b\ \ a\text{'s}}\\^{b\!-\!1 \text{ exponents}}\end{array}}\\

a\uparrow^3b&=&a\uparrow\uparrow\uparrow b\\

\vdots\end{array}$

This can create **very** big numbers if we continue developing the operators. This is an very interesting invention made by Donald Knuth, but what is it good for? I recently saw an application of $\displaystyle \uparrow\uparrow$ when reading about power towers.

Cheers! :)