There is a simple necessary and sufficient condition for a continued radical of the form $\sqrt{a_1 + \sqrt{a_2 + \dotsc}}$ to converge (where all terms $a_1, a_2$ etc. are nonnegative). Namely, that the sequence $n \mapsto a_n^{2^{-n}}$ should be bounded. This is known as Herschfeld's Convergence Theorem (though it was discovered independently of Herschfeld by Paul Wiernsberger thirty years before).

Lately, I've developed a constructive proof of this theorem, which you can find here.

It leads me to wonder: Is there any actual use for continued square roots, apart from studying them for their own sake? Googling doesn't show up anything.

A chronology of continued square roots and other continued compositions, through the year 2016by Dixon J. Jones. $\endgroup$ – Dave L Renfro Apr 28 at 18:05