$k$ is a field of characteristic $p$.

$k[t]$ has canonical first-order differential operator $\partial$

As an endomorphism of $k[t]$, $\partial^p=0$.

First way to fix it: Use the divided power differential operator $\partial^{[p]}$. Shortfall: As an endomorphism of $k[t]$, ${\partial^{[p]}}^p=0$

Second way to fix it: Use crystalline differential operators. Shortfall: No higher order operators on $k[t]$.

Question:

Is there a really big ring of differential operators which contains the divided powers $\partial^[n]$ for all $n$ and which also has a natural evaluation map to $End_k(k[t])$?