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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 only realistic answer to the title question is *no*.2012-06-01
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    Isn't the last number from **xkcd**? [[Yes, it is](http://xkcd.com/207/).]2012-06-01
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    In my humble opinion it is a bit pointless trying to get any sort of feel for the real size of numbers of these types, and better to stick with the idea that they form increasing sequences of "enormity" in which each stage is ridiculously tiny compared to the next ones in the sequence. Its a bit like trying to visualise inter-stellar and inter-galactic distances in the universe.2012-06-01
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    http://en.wikipedia.org/wiki/Graham's_number2012-06-01
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    Yes, I know of Graham's number as the question text indicates. The question is, what practical value does Graham's Number have?2012-06-01
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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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    The link does not work .2015-04-03
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    @kjetilbhalvorsen updated2015-04-16
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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