TITLE: What is a geometric, physical or other meaning of the tetration?
QUESTION [5 upvotes]: What is a geometric, physical, or other meaning of the tetration or more high hyperoperations?
Is it exist in general or it's just a math conception?

REPLY [2 votes]: $2 \uparrow \uparrow (n-1)$ is the number of sets of rank $n$. Intuitively, the rank of a set is the depth of the nesting braces needed to write it. Recursively, the rank of the empty set is zero, and the rank of a nonempty set is the one plus the maximum rank of its elements. It is also the depth of the stack needed by a pushdown automaton to recognize the braces are balanced. In short, $2 \uparrow \uparrow (n-1) = |V_n|$ where
$$V_\alpha = \bigcup_{\beta < \alpha} \mathcal{P}(V_\beta)$$