When going through with learning Grahams number, I got stuck at   $$3↑↑↑3$$  Working it through, we have  $$3↑3=3^3$$ $$3↑↑3=3^{3^3}=3↑(3↑3)$$  As such, it would appear to me that  $$3↑↑↑3=3^{3^{3^3}}=3↑(3↑(3↑3))=3↑(3↑↑)$$  Which is incorrect; the correct answer being  $$3↑↑↑3=3↑↑(3↑↑3)$$  What I'm wanting to know is where the error in the way I've worked it through, and how working $3↑↑(3↑↑3)$ through backwards to $3↑↑↑3$ would look?
I've done some research, and found this on Wikipedia. \begin{matrix}a\uparrow b=a^{b}=&amp;\underbrace {a\times a\times \dots \times a} \\&amp;b{\mbox{ copies of }}a\end{matrix} \begin{matrix}a\uparrow \uparrow b&amp;={\ ^{b}a}=&amp;\underbrace {a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}} &amp;=&amp;\underbrace {a\uparrow (a\uparrow (\dots \uparrow a))} \\&amp;&amp;b{\mbox{ copies of }}a&amp;&amp;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?
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