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[Pi/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].