Fréchet manifolds show up in some "higher geometry" situations. Here are three which I particularly like:

1.) The group of smooth loops (both based at 1 or free) in a Lie group forms a nice infinite-dimensional Lie group. These turn out to be exceptionally useful in integrable systems. For example, there is the Dorfmeister-Pedit-Wu generalized Weierstrass representation which gives an explicit parameterization of harmonic maps to symmetric spaces in terms of some holomorphic data on a loop group.

2.) Len Gross uses the Fréchet manifold of piecewise smooth paths in $\mathbb{R}^n$ to analyze the field copy problem. This is the issue that two (nonabelian) $\mathfrak{g}$-valued connections on $\mathbb{R}^n$ can have the same curvature without being gauge-equivalent. This is pretty weird; the observables of a gauge field are built out of the curvature, but with field copies you have two inequivalent fields with the same observables. At least, that's how I understand it. I'm curious if there is a tamer explanation.

3.) Brylinski does some stuff with the Fréchet manifolds of loops in a manifold in his book. I don't have it handy for reference, but there are some interesting observations. For example, the path space of a symplectic manifold is almost-complex.