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From Wikipedia, Knuth's up-arrow notation begins at exponentiation and continues through the hyperoperations:

$a \uparrow b = a^b$

$a \uparrow\uparrow b = {\ ^{b}a} = \underbrace{a^{a^{.^{.^{.^{a}}}}}}_b$ (the tetration of a and b; an exponentiation tower of a, b elements high)

This already produces numbers much larger than the number of Planck volumes in the observable universe with very small numbers; $3 \uparrow\uparrow 3$ is a relatively modest 7.6 trillion, but $3 \uparrow\uparrow 4 = 3^{7.6t} = 10^{3.6t}$.

Then there is pentation ($a\uparrow\uparrow\uparrow b = a\uparrow^3b$) and hexation ($a\uparrow\uparrow\uparrow\uparrow b = a\uparrow^4b$). The pentation of 3 and 3 is $\underbrace{3^{3^{.^{.^{.^{3}}}}}}_{\ ^{3}3}$, an exponentiation tower of 3s 7.6 trillion elements in height. Hexation is an exponentiation tower of 3s equal in height to the value of the pentation of 3 and 3. And that is just $g_1$, the first layer of calculation necessary to compute Graham's number, $g_{64}$, where $g_n = 3\uparrow^{g_{n-1}}3$.

I'm having considerable, and I hope understandable, difficulty simply wrapping my head around a number of this magnitude. So, the question is, is there value in understanding the scope of numbers produced by Knuth's up-arrow notation, or is this simply a way for mathematicians to make each others' heads explode?

If it's the latter, I leave you with the following:

$A(g_{64},g_{64})$

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    The Wikipedia article Qiaochu linked to describes a use of Graham's number. Whether you consider it practical is subjective, but it certainly serves a purpose other than merely to make mathematicians' heads explode. (If that were its only use, it's doing a pretty poor job of it: I for one haven't yet heard of any cranial-detonation-related deaths in the history of mathematics.)2012-06-01

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Yes, see "Enormous Integers in Real life" by Harvey Friedman.

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    "in real life" if you are a mathematician and you job is to think about thses problems. In which case the answer to the original question is no.2016-11-07