How far will a light beam travel? If there was an extremely strong light beam that could be seen for miles, where would the light stop? If a flashlight has low batteries, how far will its light beam travel?

Each little bit of light travels in a straight line, to the end of the universe. That's why we can see the stars. But how far away can you still see a dim flashlight? Now there are three issues:
*No light beam is perfectly collimated -- that is, to some extent it contains light that is traveling in different directions. Then as we get farther and farther away, the beam is spread out over a larger area, and is correspondingly dimmer. The stars are as bright as the sun when we are close to them, but very distant stars are hard to see. A laser makes a pretty good beam, but the best laser beam we know is a mile wide by the time it arrives at the moon. A man on the moon would not be able to see it -- it would be too dim.
*If we try to do the experiment on earth, we have to deal with the atmosphere, which is not completely transparent. A little bit of haze (small droplets of water or something), dust, bugs, or even temperature variations in the air can affect the light beam. If the light is being scattered into other directions, it is ruining the collimation. If the light is being absorbed, it's just gone.
*Finally, when we say we can't see the light, what we usually mean is that we see something else. We can't see through fog because it is bright -- it is scattering light into our eyes which blots out the light that is trying to go directly through the fog. The result is that on a foggy day, a distant building can barely be seen -- there is only a slightly different shade of gray against the sky behind it. The gray is the light the fog is sending; the slight extra brightness of the sky is the light that has made it through the fog to show us the scene beyond. This is also why most people can't see the stars at night: they are hidden behind the glow in the sky caused by the lights of the city they live in.
Putting all this together, the brightness of the light source and how well it is collimated are important, but the real answer to this question lies in factors outside the light source. The light beam doesn't stop -- it just gets harder and harder to see, and eventually disappears into the background. This is also true for flashlight having low batteries.