Except the (rather famous) example $\mathbb F_p(t)= \{ \frac{f(t)}{g(t)}:\ f,g \in \mathbb F_p[t],\ g\neq 0 \}$, which has the inseparable extension containing the one multiple root of $x^p- t$. Do we know other examples of inseparable extensions?
It seems unfortunate if we had a definition of separability only because of one example. So I suspect there are more, or, at least, there is more to this specific example.