Pulse charging, became popular after Nickel Cadmium came out, to measure the internal resistance. It did something like what you describe, but also had the advantage to help break loose (spoken loosely) materials from the electrodes (or in the case of NiCd to help prevent the formation of micro-fine shorts).
@eetcowie: "Also, Nickel Cadmium types were notorious for growing metal-fiber 'wires', that would partially short out a cell. There appeared a capacitor-based rejuvenator that would pulse a high-current into the cell to blow out the metal fibers and breathe new life."
Back in the mid-to-late 70s Popular Electronics had a simple circuit for a NiCd zapper. It had a 24V transformer and a full-wave bridge that charged a capacitor, and it had a selector switch (charge or zap), a pushbutton to zap, and a meter to monitor the charge on the cap and on the battery. A friend built one to try to resurrect some surplus NiCds he'd bought. One let out a tremendous FLASH and a ZAP, and then started to take a charge. The article said if it didn't start taking a charge after 2 or 3 zaps it never would. He was able to recover most of the "dead" batteries. Not much of a special waveform, but at the cost of used batteries, you didn't lose much by trying. I doubt that would work for Li cells!
I also read that another failure mode with NiCds was that a weak cell caught in a string of good cells would be charged in reverse.
Keep in mind a Pb-based battery needs to be kept fully charged or it dies, so it is derated to allow for capacity loss over the expected life of a battery. And in some countries you need to crank at very low temperature where you have only say 25% capacity available. In temperate climates you can get away with a much smaller battery, especially when you consider you do not have to keep the battery fully charged. It is commonly stated you need only half the capacity of a Pb-based battery. Shipping of Li is an issue, however if each cell is less than 20Wh then it is not that hard.
The Other Tesla David Blaza5 comments I find myself going to Kickstarter and Indiegogo on a regular basis these days because they have become real innovation marketplaces. As far as I'm concerned, this is where a lot of cool ...