Let $$ \Sigma_g \to E \to \Sigma_h $$ be a surface bundle over a surface. Unless otherwise stated, I'll assume $g, h \ge 2$. The theory of Thurston norm shows that surface bundles over $S^1$ often fiber in infinitely many ways (e.g. with fibers of infinitely many genera). For surface bundles over surfaces, however, the Euler characteristic (which is the product of the characteristics of the base and the fiber) provides an arithmetic constraint on the possible genera of the base and the fiber. When the genus of the base and the fiber are both at least two, this shows that there are only finitely many possible fiber genera, and an analysis of the fundamental group shows that there are only finitely many possibilities for the subgroup of $\pi_1E$ corresponding to the fiber (this follows, for instance, by work of F.E.A. Johnson). Moreover, Johnson's work also shows that when the monodromy representation $\rho: \pi_1 \Sigma_h \to \operatorname{Mod}(\Sigma_g)$ is non-injective, there are at most two fiberings, so that any bundle fibering in at least three ways must have injective monodromy.

I am aware of the example of Atiyah-Kodaira, which shows that it is possible for bundles with injective monodromy to fiber in at least two ways (of course, product bundles are a trivial example of multiply-fibered total spaces as well). However, I haven't seen any example of a bundle with three distinct fiberings when the base genus is at least two. When the base genus is one, there are trivial constructions one can do: if $M$ is any three-manifold fibering over $S^1$ in infinitely many ways, then $M\times S^1$ will fiber over $S^1\times S^1$ in infinitely many ways. I don't know of an example of a torus bundle over a surface that fibers in infinitely many ways, but I would be interested to see this, too.

With all this said, does anyone know of an example of a surface bundle over a surface of genus $h \ge 2$ that fibers in at least three ways (i.e these are pairwise non-fiberwise diffeomorphic)? What about a surface bundle over a closed $2k$-manifold of nonzero Euler characteristic that fibers in at least $k+2$ ways? (Note that a product of $k+1$ surfaces gives an example where there are $k+1$ fiberings).