I am worried that this may not be true in TOP (the topological category). Let us work in PL (piece-wise linear) instead. Every PL manifold can be expressed as a locally finite simplical complex, with at most a countable number of vertices. Thus there are at most $2^N$ of these. (Here $N$ is the natural numbers.) I think I will leave the lower bound as an exercise -- I have a way of doing this by encoding binary sequences using generalized Heegaard splittings of three-manifolds, but that is a hack. I am sure that there is a more beautiful way to give the lower bound just using non-compact surfaces.
I looked, but could not find a reference.
EDIT: Just in case the above is too brief here is an "easier" exercise. The number of simple locally finite graphs on at most a countable number of vertices is again $2^N$. [Hint: consider the adjacency matrix.] [In fact, the locally finite hypothesis is not necessary in this case.]