So LG Display is already producing flexible OLED panels, which will allow LG to produce curved phones. Now LG Chem announced that they are already mass producing stepped and curved batteries. The company is also developing bendable "cable" batteries (see image below), but it will take some years before LG Chem can produce them.
A stepped battery is a battery that is built from two battery units on top of each other in the shape of a "step". This means that LG can make a custom battery for a specific device model which will make better use of round edges, etc. The LG G2 for example uses a stepped battery (produced at LG Chem's Nanjing plant). They say this allowed them to increase battery density by 16% and add more than 3 hours to the G2 batter life. LG Chem will be able to produce designs with even 4 steps in the future.
A curved battery will be useful for a curved device such as a curved phone or smart watch. LG Chem developed (and patented) a technology which they call Stack & Folding which reduces physical stress when realizing the curved design of battery packs. LG Chem started producing curved batteries this month and these will power LG Electronics's next phone line-up, probably the curved phone we're waiting for that will use the LGD's 6" flexible OLED panel.
Cable Batteries are truly bendable and look like a cable. These are designed for wearable, bendable devices. The cable batteries do not heat up and they are waterproof.