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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
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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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