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We see addition, multiplication and exponentiation in the natural formulae that make up physics.

However, do we ever see tetration (repeated exponentiation) or higher hyper-operators in nature?

(Not counting cases where the tetration step could be trivially eliminated.)

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Tetration generally leads to extremely large numbers, which aren't very useful (except perhaps on the largest (or smallest) scales). – Dmitry Brant May 6 '13 at 14:29
This strikes me as such an arbitrary question. Blackbody radiation is proportional to $T^4$ which you could view as $T^{2^2}$ – Brandon Enright May 7 '13 at 6:28
@BrandonEnright - Edited Q to exclude trivial cases of tetration. Thanks. – billpg May 7 '13 at 13:10

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The "simple" functions you mentioned are often the result of solving "simple" (differential) equation. Repeated exponentiation is a very uncommen solution, however I do know of repeated exponentiation in formulae, often to scale dimensionless parameters.

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