The vast majority of physical theories are formulated on a spacetime that is mathematically represented by a pseudo-Riemannian manifold, i.e. a smooth manifold with a metric tensor structure. The nature of this pseudo-Riemannian manifold is very different between different theories: for example, in Newtonian mechanics it is Euclidean space (with a flat positive-definite metric); in special relativity, classical E&M, and particle physics it is usually taken to be Minkowski space (with a flat but indefinite metric), and in general relativity and cosmology it is a generic curved Lorentzian manifold. But in almost every case, the metric structure is critical to the theory.
But topological quantum field theory is an important exception to this pattern: a TQFT can be formulated on an arbitrary smooth manifold, whether or not it has a metric, because it is only sensitive to the manifold's topology (which can be defined independently of any metric).
Are there any other examples of useful/realistic physical theories that can be defined on a smooth spacetime manifold without a metric structure? In particular, any "simpler" theories, such as a theory of classical point particles or fields, whether relativistic or not? (That one might be challenging, since kinetic energy terms like $\frac{1}{2} m g_{\mu \nu} v^\mu v^\nu$ or $\frac{1}{2} g^{\mu \nu} \partial_\mu\varphi \partial_\nu \varphi$, etc. typically contain an inner product.)
(When I talk about the manifold that the theory is formulated on, I mean the physical spacetime itself, not some more abstract configuration/phase/state space.)
