$f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}}$ asymptotic?

Question

Consider for positive real $x$ :

$$ f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}} $$

How does this function behave ?

How fast does it grow ? Faster than any fixed iteration of exp sure, but How fast exactly ??

A brute estimate would be $\operatorname{tet}( \ln(x+1))$ but that would likely be a bad estimate. We have to do better.

update

$f(2) = 1,54752264961525$

Answer 1

With Mathematica there is a loglog plot

$$\text{ListPlot}\left[\text{Table}\left[\left\{x,(\log (\text{$\#$1}+1\&)\left((\log (\text{$\#$1}+1)\&)\left(\text{Fold}\left[\text{$\#$1}^{x^{\frac{1}{\text{$\#$2}}}}\&,x,\text{Range}[2,10000]\right]\right)\right)\right\},\{x,0.1,1.5,0.01\}\right]\right]$$