To start off, I was looking at the following ingeniously made form of the Gamma function:
$$\Gamma(z+1)=\lim_{n\to\infty}\frac{n!(n+1)^z}{(1+z)(2+z)\cdots(n+z)}$$
which lies on the back of
$$1=\lim_{n\to\infty}\frac{n!(n+1)^z}{(n+z)!}$$
for all integer $z$. One them multiplies through by $z!$ and use the recursive formula for the factorial to reach the above formula.
In the same light, I was wondering if a limit definition of tetration was possible. Consider the following:
$$a^{a^{a^{\dots}}}=\underbrace{a\uparrow a\uparrow a\uparrow \dots\uparrow}_nb=a\uparrow_nb$$
And then consider the following:
$$a\uparrow_nf(n)$$
Particularly, I was wondering if there was a continuous function $f:\mathbb R\to\mathbb C$ such that
$$\lim_{n\to\infty}a\uparrow_nf(n)=c$$
for some constant $c$. From there, one could imagine something like...
$$\begin{align}a\uparrow_{1/2}c&=a\uparrow_{1/2}\lim_{n\to\infty}a\uparrow_nf(n)\\&=\lim_{n\to\infty}a\uparrow_{n+1/2}f(n)\\&=\lim_{n\to\infty}a\uparrow_nf(n-\frac12)\end{align}$$
Does this seem reasonable? Does anyone know much about if this is a good path towards defining fractional ordered tetration?