Estimates on growth of $^{n}3$

Question

I was dealing with a problem on tetration and am supposed to explain why this problem was challenging to me- obviously, difficulties stemmed from the amazing growth of $^{n}3$. The question now is: Is there a way to estimate this tetration in form of "elementary" functions, as the factorial is approximated by the Stirling formula, $$\mathcal{O}\left(\sqrt{2{\pi}n}\left(\frac{n}{e}\right)^n\right).$$ Is there an elementary representation of the form $\mathcal{O}(f(n))$ for this tetration as a function of $n$?

Answer 1

By most definitions of elementary functions, the answer is no.

Every elementary $\mathbb N\to\mathbb R$ function in $n$ is bounded by ${}^kn$ for some natural $k$, since they are defined by a finite combination of operations and functions which are at most exponential.

As we have ${}^kn\ll{}^n3$, it follows that tetration grows faster than all elementary functions.