Thanks to a nudge from +Valentino Valdez, I checked out an app called +Flixel Photos. It lets you add elements of motion to still photos… basically, animated GIFs. There are other apps that do the same thing — I've also tried Cinemagram and Kinotopic, but it wasn't until Flixel that the process seemed easy and fun, rather than technical and tricky. The UI is simple, there are basic filters (plus in-app add-ons), and nice touches throughout.
A lot of apps are trying to be "Instagram for Video," and none have really stood out. Flixel falls in a sweet spot between still photos and video, and allows for the creation of both silly things and artistic things. Check it out.