  Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with
increasing Heegaard genus, and give an explicit lower bound on the Heegaard
genus of such covers as a function of their degree. Using similar methods we
prove that for any $\epsilon>0$ there exist infinitely many congruence covers
$\{M_i\}$ such that, for any $x \in M$, $M_i$ contains an embbeded ball $B_x$
(with center $x$) satisfying $\text{vol}(B_x) >
(\text{vol}(M_i))^{\tfrac{1}{4}-\epsilon}$. We get similar results in the
arithmetic non-compact case.
