TL;DR – Depends on how you define a laser beam
The other commenter is mostly right in that the answer is “forever”. However, there is some additional physics involved.
We often think of lasers as just perfect straight lines of light that come out of a device and impact a surface a long distance away. This is a good enough approximation on human scales. In reality though, due to the wave-like nature of light, even a perfectly focused beam will spread out due to a process called diffraction. To give you an idea of how severe this effect is, by the time the light from a regular handheld laser pointer reaches the Moon, the laser spot (which starts at only a few millimeters across) is larger than the Moon. This effect can be mitigated by starting with a wider beam, but the only way to get rid of it is to have an infinitely large beam to start with.
So while the light from a laser pointer does indeed go on forever, on any astronomical distance scale, the light would no longer look like a beam and would instead look fairly similar to any other light source. The power of the beam would be spread over an increasingly large area, so any detector attempting to pick up the signal would see it dim further and further. At far enough distances, the energy from the beam would be spread so thin that any detector would be receiving individual photons at a time, and beyond that point, those signal photons would arrive with more and more time between them. Eventually, the beam would be indistinguishable from the noise.
Like many things in life, the answer to “how far can a laser beam go?” is as much a question of “what counts as a laser beam” as it is anything else.
(I’ve intentionally disregarded redshift for this explanation, since that would require a more thorough explanation of frequency and quickly get overcomplicated)
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