If none of the spheres are circles, then the signature is zero by a result of Chern, Hirzebruch, and Serre that the signature is multiplicative when the fundamental group of the base acts trivially on the cohomology of the fiber.

If the sphere bundle at the last step is linear (or more generally, is assciated to a smooth disk bundle) and the disk bundle has orientable total space, the the signature is zero, because oriented boundaries have zero signature.

The above doesn't cover all possibilities, such as e.g. nonlinear smooth sphere bundles over a torus.

The total space of a fiber bundle over a sphere that isn't a circle has zero signature by a result of Chern, Hirzebruch, and Serre that the signature is multiplicative when the fundamental group of the base acts trivially on the cohomology of the fiber.