Edit: as multiple users have pointed out, the premise of my question assumes some canonical representation of real numbers as infinite nested radicals. There does not seem to be any such representation.
Khinchin's constant is the peculiar number $K$ such that for almost any real number $x$, if we write out $x$'s continued fraction representation $$x = a_0+\frac1{a_1+\frac1{\ddots}}$$ Then we have $$\lim_{n\to\infty}\sqrt[n]{a_1a_2\dots a_n} = K$$ My question begins with the fact that any real number $x$ may be written as $$x = b_0+\sqrt{b_1 + \sqrt{b_2+\dots}}$$ And, given the similarity between continued fractions and nested radicals as iterated function systems/contractions, I would think there must be some number $S$ and non-trivial function $f$ such that for almost all $x$ we have $$\lim_{n \to\infty}f(b_0,b_1 \dots b_n) = S$$ Where $f$ is probably defined independent of $b_0$.
I nervously tag this post ergodic-theory because I know Khinchin relied on it in the proof for his constant.