I find very interesting the example that if we let $Q_n$ be the maximum of n i.i.d. with distribution $U[0,\alpha]$, then $n(Q_n - \alpha)$ converges to an exponential distribution. See e.g. here for this solution.
What other examples are there that are similar in the sense that they are (1) non-trivial and (2) they don't need to rely on other theorems (e.g., like CLT or delta method), but rather can be found directly from taking the limit of the CDF.
I'm interested in both discrete and continuous examples.