If you are given a marker and a transparent water bottle partially filled with water, can you tell if the bottle is half filled or not?
Make a mark at the water level and then turn the bottle upside-down. The water level should be the same as the mark.
Lay the bottle on its side, mark the water level on both sides, then roll it 180 degrees.. if the water meets the lines, it is half full, under the lines - less than half, higher than the lines - more than half.
Inspired by Randy and his assumption of scales and more water (but not of any symmetry to the bottle), here's a solution that doesn't require a freezer or a bottle of negligible weight:
Weigh the bottle with the water in it to get the weight of bottle and water, $X$. Fill the bottle with additional water and weigh again, to get the weight of a full bottle (including water), $Y$. Empty the bottle entirely and weigh it again to get the weight of the bottle alone, $Z$. The bottle is half filled if $(Y-Z) = 2 (X-Y)$.
I'm going to assume the bottle is any shape...that is, no particular shape that happens to have some symmetry.
Therefore we can't (easily) use 'observational geometry' to transform the bottle to indicate the same water level at different positions.
So I'm left with inferring the volume.
I freeze the bottle until the water is frozen (leaving the ability to pour in more water). I fill the bottle with cold water so the frozen water doesn't melt. I pour out the water, measure it's weight, and compare to the weight of the bottle.
(Assuming the transparent water bottle is of negligible weight. What's the solution if the bottle weighs alot?)