What is the geometric, physical or other meaning of the tetration or more high hyperoperations?
Is it exists in general or it has only math concept?
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. (Something with light transmission in glass-fibers, increasing mass of a downwards rolling snowball).
Also I came once across an article called "wexzal" where the authors use the inverse of $\ ^2x$ to solve for aeroplane propulsion, and for dynamics in the explosion "chamber" of a gun-shot. (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...
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.