Brief Answer:

Nothing.

Detailed Answer:

Beyond the edge of the observable Universe there is likely just more of the same—galaxies, stars, and cosmic structures. As the universe continuously stretches out further, and all parts of the Universe get further away, we will be able to observe less and less of it.

But the real question is: “What is beyond the edge of the Universe?” Science largely agrees the Universe is self-contained. While there are interesting theories about bubble universes within a multiverse, there’s no solid evidence to suggest these universes interact with ours, and there are substantial faults in these theories. Except for what Max Tegmark, one of the foremost researchers in this area of study, labels as a level one multiverse, which really is not meaning other universes, just that our Universe is much bigger than we can observe. Even if other levels of multiverse exist, the question of an edge would remain, and the answer would be the same.

If the Universe is self-contained, then there is no edge, and beyond the Universe, there is nothing—no matter, no energy, no space—all of nature is part of the Universe, and nothing of that nature exists outside of, or beyond, it.

To help visualize this, imagine the surface of a balloon as a representation of space. Astronomers and physicists often refer to the “fabric of spacetime.”

Picture ladybugs on the surface of a balloon, representing galaxies in the universe. A better example would be with the ladybugs embedded into the surface of the balloon, but this was the best image I could find. Space is not some empty area, space is actually like the surface of the balloon, and galaxies and other things in space are embedded within that surface. As the balloon inflates, the surface stretches, causing the ladybugs (galaxies) to move farther apart. From any ladybug’s perspective, all the other ladybugs appear to be moving away, just as galaxies do in our Universe. 

From our vantage point, it may seem like we’re at the center of the Universe because everything appears to be moving away from us. However, like the balloon, there is no center. Wherever you place a ladybug on the balloon’s surface, all the other ladybugs will appear to move away from it because the space between them is expanding.

But our minds will keep trying to say, “There is a center, in the center of the balloon, and an edge of the surface of the balloon because there is air inside and outside the balloon surface.” This is partly because the balloon analogy is an attempt to explain our three-dimensional universe in just two-dimensions. We live in three-dimensions of space: length, width, and depth. But imagine living in just two dimensions: only length and width. If you were a drawing on a piece of paper, or on the balloon, that would be like living in only two dimensions. You would be unable to look “up” away from the balloon, all you could do is look and move in different directions on the balloon surface. You would have no awareness of, nor access to, anything inside or outside of the balloon surface. 

Similarly, we live embedded in the “fabric of spacetime” and can move and observe within our three dimensions of space. But, just like the balloon, there is no edge to space and nothing beyond it.

Despite the fact that there’s “no-thing” beyond the edge of the Universe, science suggests there must be something that caused the Universe—something that transcends space, time, and all the matter that makes up the Universe. However, this transcendent cause wouldn’t be found “beyond” the edge of the Universe because, simply put, there is no edge.