Look at this- $$f(a,b)=a+b$$ The next step would be to make a function $g$ such that $$g(a,b)=\underbrace{a + a + a \cdots}_{b\text{ times}}=a\cdot b$$ Then we made $h$ so that $$h(a,b)=\underbrace{a\cdot a\cdots a}_{b\text{ times}}=a^b$$
With $f,g,h$ and their inverses, we can make a lot of equations. Would we ever make or use a function $j$ such that $$j(a,b)=\underbrace{a^{a^{a^{\cdots}}}}_{b\text{ times}}$$?
The function in question is called tetration or the hyper $4$ operation, its various notations are $^ba$, or $a\uparrow\uparrow b$, or $a\uparrow^{~2}b$ $($in Knuth$)$, and it has been studied ever since the time of Euler, who discovered that infinite tetration converges for x in $\Big[e^{-e}~,~\sqrt[e]e\Big]$. More clearly, $~^{^\infty}\!\Big(\sqrt[a]a\Big)=\Big(\sqrt[a]a\Big)\uparrow\uparrow\infty=a$, for $a\in\bigg[\dfrac1e,e\bigg]$. The next operation in order is called pentation, followed by hexation, etc. Take the Greek numerals in order: tetra means four, penta is five, hexa six, etc. Their practical applications, however, are very few in number, albeit the related Lambert W function, used for expressing the solutions to $^2x=x\uparrow\uparrow2=x^x=y$, is gaining in popularity.