What is a geometric, physical or other meaning of the tetration?

Solution 1:

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,
• 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 $\ ^2x$

• 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...

Solution 2:

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.