What is a geometric, physical or other meaning of the tetration? 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?
 A: There is an entry in citizendium, where D. Kousnetzov describes his proposed general solution for the tetration-function. He links to some papers of his own where he gives more examples (there are only few so far) of physical applications.     


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*Something with light transmission in glass-fibers, 

*something with increasing mass of a downwards rolling snowball).         


Moreover once I came across an article called "wexzal" where the authors use the inverse of $\ ^2x$ 


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*to solve for aeroplane propulsion,    

*and for dynamics in the explosion "chamber" of a gun-shot.       


(For the latter see here, I made it a pdf; don't know whether this link to tetration-forum's literature-database gives open access) 
Sorry I can only give that vague hints, hope they help for a first step anyway... 
A: See Wikipedia's article on Goodstein's theorem.  Perhaps the statement that this thing is unprovable in Peano arithmetic is more interesting that its bare statement.
A: $2 \uparrow \uparrow (n-1)$ 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
$$V_\alpha = \bigcup_{\beta < \alpha} \mathcal{P}(V_\beta)$$
