How about adding option for repeating image extension to repeat image infinitely? If this option is checked, it will fill whole scene (which as we know is semi-infinite - probably there is a limit but it is very large), while retaining same x/y coordinates which now would be used to offset image (to implement things like moving clouds).
You can implement this yourself, without taking up infinite memory too! 
Yes, but the point is to make it easy for beginners. And if done properly it won’t take infinite memory either. The best example would be RPG Maker. It can include panoramas that loop and scroll infinitely and also maps that scrolls infinitely (last only RM2k/2k3 and VX Ace, not sure about regular VX).
The answer comes probably a little too late, but there’s an object in GD that does exactly that : Mosaic Object.
I use it to simulate passing fog, and it works nicely.
Are you sure you don’t downloaded extension for this? Because I don’t have it by default (have newest GD). If you downloaded extension, give me link to it.
Geez, I sure would want some extension hub where devs could upload their GD extensions. Something like GDShare, but for extensions. Perhaps even branch of GDShare.
I call it “Mosaic” because it’s called “Mosaique” in french. Maybe it’s another name in english.
Did you check your native extension when creating a new object. It should appear in the list, with Sprite and Text.
No, this is tiled sprite extension. It can repeat object several times, but not indefinitely in all directions (so when moving camera, bg will scroll). I’ve worked around this by constantly increasing width of object (height remains the same as I’m making flappy bird clone and it doesn’t scroll vertically), but it sure isn’t cleanest solution.
Can’t you use the OffsetX and OffsetY to make it “scroll” accordingly to other layers/camera?
If you are using “Center camera” range of events, that will never look right, no matter of tweaking you make.