Since we're discussing information storage rather than calculations (certainly the two are related but not the same), then per your example the information storage act would require energy to place the water molecule into the box in the first place. If you ignore that by assuming the molecule is already there, then you haven't stored anything and are simply in the intrinsic state of the box like I discussed originally. A computation with no controlled inputs yields no information, it's just nature running its course.
Perhaps you are thinking of this in a purely theoretical sense. In that case then yes, if you can harvest 100% of the energy stored when changing a value, then no additional energy is required.
But, of course, we do not live in such a perfect world. One can never achieve true 100% efficiency. And because of this, my point continues to be that it is impossible to continuously change states in order to store new information without losing some energy. Note that I never said 100% of the energy used during the original storage has to be lost. I simply said that at least some of it would be lost. In the case of the billiard balls, you only spend energy at the beginning as you mentioned, but the amount of energy required is proportional to the number of gates and, due to the laws of thermodynamics, you can never reclaim 100% of that energy after the computation is complete. Therefore, to perform a new computation and store the subsequent result, you would need to expend additional energy to reset the system and restart the computation.