Actually you can get a LOT of power out of an aluminum-air battery.
(Sorry, don't know the details of building one. Like zinc-air you're using oxygen from the air to accept electrons from the metal in a way that's non-destructive to the rest of the components, while converting the metal to its oxide.)
Aluminum is light AND aluminum oxide has the HIGHEST heat-of-formation of any compound. (It's why thermite is so energetic: The aluminum steals the oxygen from the iron, leaving elemental iron behind plus more than enough energy for the iron to be white-hot molten.) So you can both get a lot of power from a small weight battery and get it fast.
As a result, Detroit has played around with aluminum-air battery powered cars. Shovel out the bauxite every few hundred miles, replace some big electrodes every few thousand miles.
But it's really a roundabout storage battery. The way you get the aluminum in the first place is by electrolyzing aluminum oxide - again at molten temparatures - using grid power. Aluminum reduction plants are the largest single-plant consumers of grid power in existence.
Unfortunately, it's not the most efficient storage battery in the world (what with all the molten-aluminum steps between charging and discharging). So the cost isn't practical. And aluminum compounds are 'way toxic. (You get away with aluminum cans because the foods chosen to go in them don't react enough to get a lot of aluminum into YOU. But don't EVER cook something strongly acidic in an aluminum pan...)
If you have a lot of aluminum cans lying about, your recycler doesn't pay for the metal, and you have somewhere to dump bauxite, you might consider making electrodes out of them and getting back the power that went into making them. But both the smelting and the disposal of the waste give you major exposure to aluminum if you're not careful, and aluminum poisoning is permanent and has symptoms much like alzheimer's disease.
(It's also the "dialysys disease" that limited the length of time people could be on the early artificial kidneys, until they figured out that the aluminum tub in the machine was poisoning the patients and switched to something else - stainless, I think.)