This is just a curiosity and the question is really foggy. I'm wondering if there can exist a notion of "minimal smooth compactification" (when I say minimal I think something like adding a finite number of points or at least cells of dimension less than than the dimension of manifold) for a smooth noncompact manifold, in this sense: if the one point compactification of the manifold is smooth and the embedding is smooth, we are done; but what if the one point compactification is singular? Can I embed the manifold in a "minimal" compact manifold of the same dimension?

A "surface of infinite genus" $S$ is an example of a manifold that is not an open subset of a compact manifold. The reason $S$ cannot be embedded in a compact manifold is straightforward: we can find simple closed curves $a_1 , b_1 , \ldots , a_n , b_n , \ldots $ on $S$ such that, for each positive integer $g$, the curves $a_1 , b_1 , \ldots , a_g , b_g$ form a standard basis for a surface of genus $g$. Thus, considering the product in homology of the classes of these curves, we deduce that they are independent. If $S$ were an open subset of a compact manifold $M$, the same argument would imply that the images of the curves constructed above would also be independent in the homology of $M$. This contradicts the fact that the homology of the compact manifold $M$ is finite dimensional. Observe that this example is not particularly different from the example of the complement in the complex plane of the integers. Indeed, $S$ can be realized as the double cover of $\mathbb{C}$ branched along the integers. 


What do you want to do with an open annulus in the plane? Already open subsets of the plane may need infinitely many points added to compactify them in a sensible way. 

