An easy upper bound for $S^*_N$ is $\lceil N(N-1)/4 \rceil$, the integer ceiling of half the length of a legal sequence taking $[1,\ldots,N]$ to $[N,\ldots, 1]$. If some transposition $(i, i+1)$ occurred more often in the sequence, there would be two $(i, i+1)$ in a row which would cancel each other out / cannot happen in a legal sequence.

A small example shows that this bound is sharp. In the [Dauvergne paper](https://arxiv.org/abs/1802.08934) Sam mentioned in the comments, Figure 5 on p8 shows the permutahedron, a visualization of the $N = 4$ case.  By Stanley's formula (1) on p3, there are 16 legal paths from $[1,2,3,4]$ at the bottom to $[4,3,2,1]$ on the top, each of length 6.  Some of these paths have two transpositions each of $(1,2)$, $(2,3)$, and $(3,4)$, while some have an unequal distribution.  The one along the right-hand side of the figure, for example, is red-green-red-blue-green-red, i.e., $(3,4),(2,3),(3,4),(1,2),(2,3),(3,4)$.  By the reasoning above, no legal path / reduced word could have a certain transposition in 4 of the 6 steps, so $S^*_4 = 3$.  The fussiness about the integer ceiling comes from the easier example of $N=3$ where the only possibilities are $(1,2),(2,3),(1,2)$ and $(2,3),(1,2),(2,3)$ which shows $S^*_3 = 2 = \lceil 3/2 \rceil$.