Let $K$ be a field, $A$ be a square matrix with coefficients in $K$ and $\lambda\in K$. Usually, people call the algebraic multiplicity of $\lambda$ to the the biggest integer $a\ge 0$ such that $(X-\lambda)^a$ divides the characteristic polynomial.
Is there a name of the biggest integer $m\ge 0$ such that $(X-\lambda)^m$ divides the minimal polynomial of $A$ ?
This number seems to be indeed interesting.
Remark: People usually call geometric multiplicity the dimension of the eigenspace associated to $\lambda$. Writing this $g$, we find $$g\le a, m\le a$$ but we cannot really compare $g$ with $m$: both inequalities $g\le m$ or $m\le g$ are possible. Looking at the Jordan form (by extending the scalars if needed), then $a$ is the sum of the sizes of blocks with eigenvalues $\lambda$, $g$ is the number of blocks and $m$ is the size of the biggest block.