Let $M$ be a smooth manifold. Is it always possible to find a manifold $N$, compact, and a diffeomorphism of $M$ onto an open subset of $N$?
I know that if $M$ is given some Riemannian metric, then it may be "non-extendible." However, I don't understand any examples of this phenomenon, so I cannot judge the extent to which it cares about the metric.
On
No, it's not always possible. Consider the "orientable surface of infinite genus": let $S$ be a torus with two small disks removed, and consider the colimit of $$S \to S \# S \to S \# S \# S \to \dots$$ where the connected sum means that you glue along the disks that you have removed. (Imagine a surface with an infinite number of holes, basically.) Then you can check the classification of closed surfaces to see that this isn't homeomorphic to any open subset of any of them.
While in the category of topological spaces you have always Alexandrov compactification (and other compactification schemes also), in the context of manifolds, it appears that the answer is no.