Choose positive integers $D$ and $N$. Roll a fair $D$-sided die $N$ times, recording the number of times each of the $D$ outcomes are rolled, say $r_1, r_2, \ldots, r_D.$ What are the asymptotics of $\min(r_1, r_2, \ldots, r_D)$?
They're not independent, so I can't directly apply the Fisher–Tippett–Gnedenko theorem but that's probably a good starting point.
If it makes it easier you can assume $N \gg D.$ Of course the leading term is $N/D$ but what’s the second-order term? Maybe $\asymp \sqrt{N/D}$?