The $n$th tetration of $a$, most often written $^na$, is defined as $n$ copies of $a$ combined by exponentiation in the style of . 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.
• 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.