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Is there a term (or operation) for repeated exponentiation?

I.e. Repeated addition is multiplication, repeated multiplication is exponentiation, repeated exponentiation is X?

Is there even such a term for such a "higher order" operation, seeing as we could potentially deal with this by increasing the exponent?

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    $\begingroup$ Search for tetration $\endgroup$ – Claude Leibovici Jan 19 at 9:11
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The word you are looking for is probably tetration

at least if you are looking for something like $$\underbrace{a^{(a^{(\cdot^{\cdot^{a)\cdots)}}}}}_n$$

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