Is there a way to create a seamless image in the compositor? Can a rendered frame be made seamless, by some compositor tricks? What I'm looking for is something similar to the GIMP filter Make seamless.  

Here's an example
Original (downloaded from Pixabay):
  

After Make seamless applied:
  

I could, of course, do this in GIMP, but I'd like to avoid post processing in external applications, and get a ready made image directly out of Blender. My goal is to have a repeating background and composite non-repeating text on top of it, and I'd like to be able to do it with two render layers or, if necessary, two scenes in the same .blend.  

When searching Blender.SE, I found several questions about seamless images, but they all seemed to want to know how to use pre-existing seamless images, or how to get a texture to repeat seamlessly on an object. The ones I actually opened, did not answer my question, since I'm aiming to create a seamless image from a render.
The question depends on the nature of the source image. In the Fireworks example the edge is already seamless (black). So you can use the compositor. Otherwise you would prepare the edges as the other answer suggests with a blending mask.


Import image
Add a Translate node, turn on "Wrap Both Axes"
Alter X and Y offset
You can add the original image back over the top with an "Add" mix
node.