The short answer is: probably nothing much, because galaxies are very fuzzy objects without "edges".
If you look at the stellar disk, it just fades out in density, with no evidence for a sharp cutoff (this is true of disk galaxies in general). There may or may not be a "break" at some radius, but this just marks the beginning of a steeper falloff in density, not an "edge" in any real sense. The gas disk extends even further out (that is, further out than we can easily trace the stars in the disk), and doesn't show much evidence for a sharp cutoff either.
There's also a round halo of stars, which is low density but extends to at least 400,000 light years; two stars were recently discovered which are plausibly part of the halo and located at distances of about 800,000 light years. (Note that there are other galaxies -- such as the Large and Small Magellanic Clouds -- which are closer to us and to the center of the Milky Way, so you have to accept that individual galaxies can be found "inside" other galaxies.)
And then there's the halo of dark matter...
I can think of two ways in which you might talk about the outer boundary of an isolated galaxy (which the Milky Way sort of is) from a theoretical perspective. The first is the idea of a halo "accretion shock", in which relatively cold gas falling onto the galaxy from intergalactic space runs into the galaxy's halo of hot gas ("hot" = millions of degrees), creating a shock where the infalling gas slows down and the local density of gas goes up. (This is a little bit analogous to the heliopause, though the direction of gas flow is the other way around: the heliopause is where the outflowing solar wind shocks into the more or less stationary interstellar gas.) The problem is that extensive, hot halos of gas are thought to form only around massive galaxies, and it's not clear the Milky Way is massive enough to qualify. It also won't work for galaxies in massive groups and clusters, where the individual galaxies aren't accreting gas any more (the cluster is what has the hot halo and the accretion shock).
The other possibility is a region where the infalling dark matter particles merge into the dark-matter halo (whose particles are orbiting within their common gravitational potential, and so are not as a whole falling towards the center any more). Simulations suggest that as one moves inward the local density of dark matter starts to increase rather rapidly at this point (and the average velocity of the dark matter particles changes), but it's still a fairly smooth process (unlike gas, the dark matter particles aren't "shocking" into each other). There's no way of determining where this is for the Milky Way, other than something really vague like "maybe 1 million light years"?