Nested radicals

Given a list of integers acting like digits one can consider representing numbers in the form Fold[Sqrt[#1 + #2]&, 0, Reverse[list]]. A sequence of identical digits d then corresponds to the number (1 + Sqrt[4d + 1])/2. (Note that Nest[Sqrt[# + 2] &, 0, n] 2 Cos[π/2^{n + 1}].) Repeats of a digit block b give numbers that solve Fold[(#1^{2} - #2) &, x, b] x. It appears that digits 0, 1, 2 are sufficient to represent uniquely all numbers between 1 and 2. For any number x the first n digits are given by

Ceiling[NestList[(2 - Mod[-#, 1])^{2} &, x^{2}, n - 1] - 2]

Even rational numbers such as 3/2 do not yield simple digit sequences. For random x, digits 0, 1, 2 appear to occur with limiting frequencies Sqrt[2 + d] - Sqrt[1 + d].