I can't say I've personally implemented this, but here is my suggestion:
With a sufficiently large background "world" and a sufficiently narrow field of view on that world, you could implement a "inner ring" background world similar to the world in the Halo games. Put the background camera inside the ring, pointing "outward" towards your background world, and move it along the inner orbit. Render this to texture, and use that as your background.
Of course you could also put the ship, enemies, etc inside the ring itself to avoid the render to texture process, but assuming an Ikaruga-like game, having to account for the curve of the ring (i.e. a third dimension, and at least one constantly changing axis) might cost more than the RTT, and would certainly be harder from a coding standpoint.
With clever maths, you can probably turn a "ribbon" world of known dimensions into a ring to make level editing easier. Any moving items in the background will need extra attention though.
Note the caveats, though. Without a large enough ring or with too wide a camera FOV, things will probably look weird, although perhaps you could account for that with some other method, like fish-eye.
EDIT: Thinking about the particular case of repeated segments that eventually change, I think maybe two half-rings would be a better solution. A repeatable half-ring would be able to connect with itself seamlessly, and when you're done repeating, swap out the next half-ring for the next segment. This is a little more complicated than a single, giant ribbon that gets turned into a ring, but if you need to repeat on-the-fly instead of a pre-specified number of times, it's a potential solution.
EDIT 2: For situations where you need to be able to move and repeat on both axes, consider a similar approach but with a torus. Could be much more challenging to construct.