I am worried that this may not be true in the topological category.  Let us work in PL instead.  There are only countably many compact PL manifolds.  Every non-compact PL manifold can be expressed as a locally finite simplical complex.  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 in dimension three, but it 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.