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$\DeclareMathOperator\PSL{PSL}$Let $G_1,...,G_m$ be pairwise non-isomorphic maximal lattices in isometry groups of real-hyperbolic spaces $\mathbb H^{k_i}$ (possibly of different dimensions $k_i\ge 3$) with nontrivial abelianizations. (Note that every lattice is contained in a maximal lattice as a finite index subgroup.) For instance, we can take some pairwise noncommensurable arithmetic subgroups of the simplest type which do not preserve orientation: All such lattices have nontrivial homomorphisms to $\mathbb Z_2$. Take $$ G=G_1\times ... \times G_m. $$ Suppose that we have an inclusion $G\to H$ as a finite index subgroup. Then, by Mostow Rigidity, the group $H$ acts isometrically (possibly, with finite kernel) on the product of hyperbolic spaces $$ X=\mathbb H^{k_1}\times ... \times \mathbb H^{k_m} $$ extending the product action of $G$. In view of maximality of each $G_i$ (and the assumption that these groups are pairwise nonisomorphic), this action of $H$ cannot permute the factors and, thus, again, by maximality, the action has to be equal to the action of $G_i$ on the corresponding factor in the product. Thus, $H$ is an extension of $G$ by a finite subgroup $K$, the kernel of the action of $H$ on $X$. The group $G$ and, hence, $H$, has a surjective homomorphism to a product of $m$ nontrivial abelian groups (abelianizations of the groups $G_i$). Thus, the rank of $H$ (the minimal number of generators) is at least $m$. The same argument works when we allow some $k_i=2$ as long as the corresponding group $G_i$ is a triangle reflection group.