This is encompassed by some of the fancier examples above, but a good "classic" example of natural infinite dimensional manifolds is the space of paths connecting two points on a finite-dimensional base manifold. For example a natural interpretation of deriving the Euler equations from the Calculus of Variations is to find points in the path manifold where the derivative of the action functional vanishes. If you use the length functional you get the geodesic equations of a Riemannian manifold. Going a step further and studying the second derivative in the path manifold leads to some nice theorems relating the curvature of the base manifold and its topology.