What is the maximum total number of zeroes a univariate polynomial $f\in\mathbb{C}[z]$ of degree $d$, together with all of its derivatives, can have at $k$ given points of $\mathbb{C}$?

I am interested in this question for constant $k$, say $k=3$.

One trivial bound is $d \choose 2$, an other is $kd$.

Does there exist an example of a polynomial with more than $k+d$ zeroes in total of it and its derivatives?

In other words, if $p$ is a polynomial and $x_1,\ldots x_k$ are his zeroes then I am asking how many $i,j$ there exist such that $p^{(i)}(x_j)=0$