The reason why $e$ is natural is that $e^x$ grows exactly at the rate of its own value. This may seem $e$ has some special place in maths, but in fact, every exponential function grows proportionally to its own value. The number $e$ just happens to be the only one to achieve the perfect coefficient-1 with the help of its infinitely-many fine-tuned digits, just like how $\pi$ achieves the perfect radius-1 curvature.

It’s very hard to imagine an exponential index that is not an integer. How on earth can an exponential index grow continuously? Things can only multiply by themselves once, twice, but not 1.5 times. We may imagine it using roots, but this is not intuitive.

In primary school, we also took some time to understand what multiplying by a decimal number means — addition in fractions. Exponentiation follows this logic but one order higher — multiplication in roots.

This got me thinking: can we do this one more time? Can we go beyond exponentiation? This is the fourth-order operation: tetration.

Tetration is a tower of powers. For example, Tetration(2) becomes 2^2, and Tetration(3) becomes 3^3^3. Each number becomes a tower of powers with it’s height the same as itself.

Is there a special number in the realm of tetration, like e? Maybe the number whose “tetrational rate” (whatever that means) is equal to the value of the function itself. Wait, if we can define this rate, maybe we can even futher define a higher-order differentiation operation!

To make this happen, we need to extend the definition of tetration to real numbers. Unfortunately, this steps into the realm of unsolved math problems. There’s no generally accepted function to extend the definition of tetration to real numbers. If tetration is not even a continuous function, there’s no hope in finding its derivative.

To make things worse, unlike exponentiation, which models many natural phenomena aptly, tetration is a completely useless operation because it grows so fast that we run out of atoms in the universe by the time we reach the number 4. Nothing in the universe can be modeled using tetration. I can’t even think of an analogy to demonstrate its growth rate. this is because in natural phenomenon, exponential index is usually a constant number. If you think of it as a iterative process, in exponential function, the result of the last iteration is plugged into the base, while the exponent remains unchanged. in tetration, the result is plugged into the exponent while the base remains unchanged.