Reinventing the battery

A battery is an electrochemical cell that, for all intents and purposes, stores electricity. Important, because electricity cannot be stored in its original form. Electricity is movement, and storing movement for later is nonsense.
It has become apparent that we need better batteries. More and more of our civilization depends on portable electronic devices, and electricity is a very good energy source, but not very portable. A better battery would mean not only longer lasting laptops, smartphones, and radios, but also would improve the plausibility of perfectly clean electric cars.
It's a matter of chemistry. The ideal battery would have high energy density, a reversible reaction (so that you can recharge it at the nearest outlet), inexpensive materials (because we need a bajillion of them, and the less they cost, the better), secondary reactions (so that two energy can be extracted from both, thereby extending running time), and be safe. (Exploding gadgets would be terrible!)
Unfortunately, I don't know enough about chemistry to help. Only the basics.