My question is, what kinds of batteries have the highest amount of watt hours per cm3? I know, comparatively, that for example li-ion batteries in phones generally have a higher capacity than, say, duracell rechargeable aa batteries. Preferably, I would prefer each individual battery to be under 5 volts, and not cost more than a hundred dollars.
Batteries do not exist on a single line that measures say capacity. They exist in a "phase space" that includes capacity, robustness (how they put up with overcharge or overdischarge) and safety (ask Samsung). These are examples there may be more.
You have given the answer to your question the Li-ion batteries are by far the highest in pure capacity. However you need to take into account the other dimensions to get a really good answer.