For a while, I've been wondering why this pattern seems to allude to the fact that $0$ is one of the inverses of exponentiation.
\begin{align} & x \cdot -1 = -x & \text{Inverse of addition} \\ & x^{-1} = \frac{1}{x} & \text{Inverse of multiplication} \\ & ^{-1}x = n & \text{Inverse of exponentiation} \end{align} where $^{n}x$ indicates tetration on $x$. It seems that according to Wikipedia (not a great source, I know) $^{-1}x$ is equal to $0$ https://en.wikipedia.org/wiki/Tetration#Negative_heights, which would imply that $n$ (the exponential inverse) should be $0$. At first this didn't make sense to me, but I quickly realized that this makes sense that it seems that any number and its inverse for some operation, when combined under that operation, produce that operation's identity, e.g. $x+-x=0$ and $x \cdot \frac{1}{x} = 1$. Assuming that $1$ is the exponential identity, it makes perfect sense that $0$ is the exponential inverse as $x^{0} = 1$.
Some things to note about this defintion of the exponential inverse is that $0^x \ne 1$, but instead $1^x = 1$. I currently don't see how this is justified by this technique. Also, it doesn't seem that $0$ is tied to either of the exponential inverse functions in the same way $\frac{1}{x}$ is tied to division and $-x$ is tied to subtraction, though I would expect to be tied to the $n$th-root operation.
I'm sorry if my explanations are a bit vague. I am not very knowledgeable with hyperoperations and this area of math. Please let me know if I need to clarify anything.