Melting down and casting metals is incredibly satisfying, and bootstraps a whole pile of other possibilities — including ones that can generate an income. It also puts you on (or further down) the path to self-sufficiency.
At the very least, melting down aluminium cans will let you reduce the volume of cans stored drastically. That will buy you some time whilst you are thinking about what to do with the metal that is accumulating.
Unless you have something specific in mind, I would suggest casting them into cylinders — of various diameters — that can be efficiently stacked/stored, and which could then be placed onto a lathe. You can fashion washers, pins, bolts, discs, knobs, axles, barrels, and a myriad of other small components that can be used to make or repair things.
If you're not interested in lathe work, then cast into blocks and thick plates. These can be ground/sanded/machined down into a variety of larger, more "structural" elements — which can be welded/brazed with or bolted to other components you either make or buy.
Regardless of whether your stock is cylinders or blocks, you can always get creative and re-cast them. 3D printers are super-cheap nowadays. Use one to print off more artistic or complex things (gears and the connectors for geodesic domes come to mind). Use the 3D-printed piece as a "master" to create the sand casts that you pour your aluminium into.
Once it dawns upon you that you can shape metal into almost anything, your problem turns into one of having a lack of time to do all the ideas that start flooding into you your head. :)
Even if you never actually end up making anything, you can still exchange the metal at any point in time for cold, hard cash. Might prove useful if the times ahead ever get rough.
Finally, keep in mind that if your apartment complex has stopped its recycling program, then you have a lot of neighbours with exactly the same "problem" as you do. Instead of thinking about what you could do with (say) 1kg of aluminium a month, maybe you should be thinking about what you could do with 10 or 20 times that amount? Efficiency increases, and waste decreases, as you scale up the operations.