Tetration is iterated exponentiation, just as exponentiation is iterated multiplication. It is the fourth hyperoperation and often produces power towers that evaluate to very large numbers.

The $n$th tetration of $a$, most often written $^na$, is defined as $n$ copies of $a$ combined by exponentiation in the style of power-towers. The evaluation proceeds right-to-left as is the norm for nested exponentials:
$$^na=\underbrace{a^{a^{\ \dots\ ^{a}}}}_n$$
This classical definition works whenever $a$ (the *base*) is an integer or positive real number and $n$ (the *height*) is a non-negative integer. Alternative notations include $a\uparrow\uparrow n$ (Knuth's), $a\to n\to2$ (Conway's) and the text notation `a^^n`

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The alternative name *hyper-4* for tetration reflects its place as the fourth hyperoperation after addition, multiplication and exponentiation. There are three main types of questions relating to tetration:

- If the base is a natural number, the result of tetration will be a very large natural number and number-theoretical questions like "What is $^na\bmod N$?" are relevant. For example, computing the last digits of Graham's number involves computing the last digits of $^n3$ for sufficiently large $n$.
- How can tetration be extended beyond the classical definition? Complex bases can be easily accommodated, while the extension to infinite heights ($^\infty z$) features a connection to the Lambert W function: $$^\infty z=\frac{W(-\ln z)}{-\ln z}$$ In contrast, there are several proposed extensions of tetration to real or complex heights, but none have been widely accepted.
- How can tetration be reversed? Just as the two inverses of exponentiation are roots and logarithms, the two inverses of tetration are
*superroots*and*superlogarithms*. As with tetration's extensions, there are many open questions relating to these two inverses.