  In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph $G$
and a positive integer $k$, and the objective is to decide whether $G$ contains
a minimal vertex cover of size at least $k$. This problem has been considered
in several articles in the last years. We focus on its kernelization, which had
been almost unexplored so far. We prove that MMVC does not admit polynomial
kernels parameterized by the size of a minimum vertex cover, even on bipartite
graphs, unless ${\sf NP} \subseteq {\sf coNP} / {\sf poly}$. Motivated by a
question of Boria et al. [Discret. Appl. Math. 2015] about the existence of
subquadratic kernels for MMVC parameterized by $k$, we rule out their existence
unless $P=NP$, if we restrict the kernelization algorithms to apply only a type
of natural reduction rules that we call "large optimal preserving rules". In
particular, these rules contain the typical reduction rules to obtain linear
kernels for Vertex Cover. On the positive side, we provide subquadratic kernels
on $H$-free graphs for several graphs $H$, such as the bull, the paw, or the
complete graphs, by making use of the Erd\H{o}s-Hajnal property in order to
find an appropriate decomposition.
