Consider the ring of polynomials over a field or ring: $R[x]$. We can define a formal differentiation operator: $D$ which maps $x^n$ to $nx^{n-1}$ etc.
For any particular element of $R[x]$, sufficient applications of $D$ will result in $0$. However, there is no $n$ such that $D^n$ maps all elements to $0$.
So, do we call $D$ nilpotent? If we do then do we say that it is nilpotent with infinite degree? If we don't then is there another standard term?
(Not homework, just an old man trying to exercise his ageing brain.)
We say that $D$ is locally nilpotent.