Estimate a covergent (?) infinitely nested radical

first of all, let's determine if it converges. We can see, the nested radical can be written as an infinite product.

$\prod_{n=1}^\infty\sqrt[2^{n}]{\sum_{x=1}^nx^{x}}$

An infinite product

$\prod_{n=1}^\infty a_{n}$

converges if

$\lim_{n \rightarrow \infty}a_{n}=1$ (at least for nested radicals).

In this case, we need

$\lim_{n \rightarrow \infty}\sqrt[2^{n}]{\sum_{x=1}^nx^{x}}=1$

in that sum, the highest value is for $x=n$ and all the other terms are negligible, so,

$\sqrt[2^{n}]{\sum_{x=1}^nx^{x}}\sim\sqrt[2^{n}]{n^{n}}$

for $n\rightarrow\infty$

This is the way I solved this limit

$\lim_{n \rightarrow \infty}(n^{n})^{\frac{1}{2^{n}}}=\lim_{n \rightarrow \infty}n^{\frac{n}{2^{n}}}=\lim_{n \rightarrow \infty}e^{\frac{n\ln (n)}{2^{n}}}=e^{0}=1$

So, now we know the product converges, let's see how to estimate the value.

let's consider the first few terms of the nested radical

$1; 5; 32; 288; 3413; 50069$

so, the value of the radical (k)

$k>\sqrt[4]{5}*\sqrt[8]{32}*\sqrt[16]{288}*\sqrt[32]{3413}*\sqrt[64]{50069}$

we can compare each of these to, for example, $\sqrt[n]{2}$ so,

$\sqrt[4]{5}>\sqrt{2}$

$\sqrt[8]{32}>\sqrt{2}$

$\sqrt[16]{288}>\sqrt{2}$

$\sqrt[32]{3413}>\sqrt[3]{2}$

$\sqrt[64]{50069}>\sqrt[5]{2}$

and than

$4<\sqrt{2}*\sqrt{2}*\sqrt{2}*\sqrt[3]{2}*\sqrt[5]{2}<k$

from the inequalities above, we can expect an exponential growth of $n$ in $\sqrt[n]{2}$ , like

$\sqrt{2};\sqrt{2};\sqrt{2};\sqrt[3]{2};\sqrt[5]{2};\sqrt[7]{2};\sqrt[11]{2};\sqrt[18]{2};\sqrt[31]{2};\sqrt[54]{2};\sqrt[96]{2};...$

To estimate a maximum value we can try to slow down the exponential growth, like

$\sqrt{2};\sqrt{2};\sqrt{2};\sqrt{2};\sqrt[3]{2};\sqrt[5]{2};\sqrt[7]{2};\sqrt[10]{2};\sqrt[14]{2};\sqrt[20]{2};\sqrt[27]{2};\sqrt[38]{2};\sqrt[56]{2};...$

and now we can say

$4<k<\sqrt{2}*\sqrt{2}*\sqrt{2}*\sqrt{2}*\sqrt[3]{2}*\sqrt[5]{2}*\sqrt[7]{2}*\sqrt[10]{2}*\sqrt[14]{2}*\sqrt[20]{2}*\sqrt[27]{2}*\sqrt[38]{2}*\sqrt[56]{2}<8$

and finally

$\boxed{4<k<8}$

We can check with a calculator, to find out that the value of the nested radical is $6.52306176...$