Rich asks: << Do all such sans-0-digit strings appear somewhere in the square of a sans-0-digit number? Statistically, it seems unlikely that most such strings appear in the square of a *shorter* s0d number. This would mean that, if all s0d numbers occur in the closure of {2}, most of them are reached through a path that involves shrinking down from a larger number. Second question: Do all the s0d numbers generate the same set? Or, Do all s0d numbers generate 2?
Thinking of a digit string s as defining s/10^p in [0,1), it's clear that by going out enough places the distance between successive squares (k+1)^2/10^2r - k^2/10^2r = (2k+1)/10^2r can be made small enough so that there exist r = r(p) sufficiently large and k = k(p,r) such that .s < k^2/10^2r < .s + 1/10^p, ensuring that s occurs as a substring of k^2. --Da