The LG Flex actually does, though we’re not quite sure why

http://www.youtube.com/watch?v=LahYuSyWidc

We first heard the LG G Flex – a smartphone with a curved OLED display – was on its way just over a month ago. That was followed by renders and then photos and video. What none of these showed, however, was the the flexible display really does flex. In fact, the whole phone does … 

Like Samsung’s Galaxy Round, which is essentially a prototype put on limited sale in Korea only, we’re not sure that the handset will ever make it to the U.S. We’re also not entirely sure what the benefit of the flexing might be, though it does suggest that it may be more robust than a conventional rigid design.

Engadget reports that the handset also has a ‘self-healing coating’ which can repair scratches on the rear of the casing.

Second video below.

https://www.youtube.com/watch?v=H56j16OhvbM