  We study relations among characteristic classes of smooth manifold bundles
with highly-connected fibers. For bundles with fiber the connected sum of $g$
copies of a product of spheres $S^d \times S^d$ and an odd $d$, we find
numerous algebraic relations among the so-called "generalized
Miller-Morita-Mumford classes". For all $g > 1$, we show that these infinitely
many classes are algebraically generated by a finite subset.
  Our results contrast with the fact that there are no algebraic relations
among these classes in a range of cohomological degrees that grows linearly
with $g$, according to recent homological stability results. In the case of
surface bundles ($d=1$), our approach recovers some previously known results
about the structure of the classical "tautological ring", as introduced by
Mumford, using only the tools of algebraic topology.
