Clearly the title needs clarifying. Allow me to let "how many" to mean a set larger than a skeleton of the category of Fréchet manifolds and smooth maps, if this category is indeed essentially small.
Since I suspect this category is not essentially small, I would be satisfied with knowing the answer to the question with "Fréchet manifold" replaced by "Fréchet manifold admitting smooth partitions of unity" (a relevant related question is Which Fréchet manifolds have a smooth partition of unity?). I would be likewise be happy to know it for metrisable Fréchet manifolds (which I believe to include those with smooth partitions of unity).
EDIT : I will add that my motivation is to see whether there is a sensible essentially small category of Fréchet manifolds including smooth mapping spaces, closed under natural operations (finite limits, when they exist, and reasonably-sized coproducts), which is defined by topological criteria not explicitly depending on cardinals. An instance of the latter would be something like asking that the underlying set is smaller than some given cardinal