A product related to logarithms?

Question

Given integer $n$ consider the sequence

$$t_{i+1}=\log\frac{\log n}{t_i}$$ where $t_0=2$.

What is a good estimate of the product $$\prod_{i=0}^pt_i$$ where $1\leq t_p\leq2\leq t_{p-1}$ holds?

Answer 1

If $n<\exp(2e)$, then $t_1<1$ and all subseqent $t_i < 2$ (if they exist, the series may cut off at some point), so $p$ is not defined.

If $\exp(2e) < n < \exp(2\exp(2))$ you can easily check that already $1 < t_1 < 2$, so $p=1$ and $$\prod_{i=1}^p t_i = t_0t_1 = 2 \log\frac{\log n}{\log 2}$$

Finally, if $n > \exp(2\exp(2))$ it can be proved that $t_i > 2$ for all $i>0$, so again $p$ is not defined.