>...Power. Batteries more specifically. We just don't have a way to power anything for very long that doesn't have a cable attached to a wall attached to a power station...
To be clear (not to be snide, just to improve clarity), increased power density is the ability to discharge energy in larger amounts in shorter time periods (think heavy lift olympians). Energy density is the ability to store large amounts of energy for longer periods of time (think long distance runner). Some applications, like cell phones are quite happy with high energy density, but relatively low power density. Others, like an battery-electric drill, need more power density but can offset the low energy density by swapping out batteries and having a rotable pull.
Great idealic batteries for this application would have both large energy and power desnity. I say this because people often overlook "batteries" that people don't refer to ass batteries because they generally don't come in a self-contained cell (they have subsystems that don't benefit much from cellular architectures). Case in point are Hydrogen fuel cells. I'm not saying their better than Li-ion, but they are for certain applications. Often H2 fuel cells are paired with Li-ion batteries or supper capacitors to provide improved power when transient energy bursts are needed. H2 fuel is stored in some tank - perhaps a pressure tank, perhaps in the form of Aluminum and water reaction, or for this application it may likely be some metal hydride that stores H2 'benignly'. However, to your point, even these metal hydrides have a lot of room and potential for improving energy density.
My only point here is that, cureently, the best answer may not be a single cellular battery, but some hybriid of energy storage technologies.