Apple has busy patenting several new OLED and flexible display patents lately. In June 2012 the company filed for a new patent in Europe that describes several new technologies for a future iPhone device - mainly a wraparound flexible transparent display that enables new form factors, aided with an alumina powder liquid-metal process.
The main invention is an electronic device that has a 'hollow display cover structure'. The device is like a long tube (or cylinder, or other shapes) that has a wraparound transparent flexible display. Apple also describes some applications that can take advantage of these new displays and designs - a stock ticker, gaming, news, and more.
This is not the first time Apple patents a device with a wraparound display. In March 2013 the USPTO unveiled a patent that also shows such a device. That patent also describes an option to include multiple transparent displays, overlaid on top of one another.
We're still waiting for Apple to release their first OLED products. Last month Bloomberg posted that Apple's upcoming new iPhones will use a displays with "glass that curves downward at the edges". It's highly unlikely that Apple could use a flexible OLED next year as production capacity for these screens (at both SDC and LGD) is still very small. Reports from Korea suggest that LG Display is very close to striking a deal with Apple to supply flexible OLEDs for the iWatch, which is more realistic as the limited screen size means that LGD may have enough capacity next year.