Let $S$ be the set of functions $f:\mathbb{R}\to \mathbb{R}$ such that $\sqrt{f(1)+\sqrt{f(2)+\sqrt{f(3)+\dots}}}$ converges.
A function $q(x)$ dominates $p(x)$ if there exist an m such that $q(x)\gt p(x)$ for all $x\gt m$.
Take all functions $f(x)$ from $S$ and put $O(f(x))$ in $S2$.
Which function $g(x)$ in $S2$ dominates all others?
Are there asymptotic lower and upper bounds on $g(x)$?
Herschfeld proved that the nested radical converges iff $f_n^{2^{-n}}$ is bounded. In view of this, you consider bounded sequences under the same domination relation. It is clear that the increasing sequence of all integral constant sequences is cofinal (bounds every sequence). Coming back to the original problem, the following sequence is increasing and cofinal: $$ \begin{align*} &1,1,1,1,\ldots \\ &2,4,16,256,\ldots \\ &3,9,81,6561,\ldots \\ &4,16,256,65536,\ldots \\ &5,25,625,390625,\ldots \\ &\ldots \end{align*} $$ The $k$th sequence starts with $k$ and continues by repeated squaring.