Many apps, OSes, and sites have "flat" design, but what designs and design languages might have influenced Apple's direction with iOS 7, or at least predate it while seeming to reflect the same sort of thinking?
One app which seems strikingly prescient, in the sense that one could say iOS 7 emphatically ratifies its design, is Rdio. When the redesign launched, I remember initially thinking: My goodness, look at all this white space and text! I wasn't bright enough to see what problems the design solved for a cross-platform, multi-device service like Rdio, one which needed to develop across UIs and OSes very quickly.
(If I were to attempt a defense: I think the text layout looked slightly messy at first. But I'm not sure, and it's safe to say I was simply totally wrong, as more than one friend told me at the time).
When you think of the awful Spotify iOS apps, which are chrome-heavy, awkward, feature many long flows for what should be simple actions, and above all look so different from the web, Mac, and other versions of Spotify, you can immediately see how Rdio's solutions were quite smart. Spotify has siloed app UXes; Rdio is consistent, seems easier to iterate upon across platforms, and gives users a sense of familiarity.
(Also: Rdio doesn't make you choose which kind of track data —title, article, album— to search for before you search, before you even know what their catalog has; that Spotify does this seems berserk to me, but I'm sure they have their reasons).
Rdio on iOS and the web has lovely background blurring to give one an immersive sense of the aesthetic of whatever one is listening to. As it happens, background blurring of this sort (and with similar effects) is everywhere in iOS 7:
Rdio is also "flat," but not just to be faddish; instead it is to do what Apple asserts iOS 7 does: prioritize content and then get out of the user's way. Color in the UI often denotes action. There are fewer buttons for touch-targeting, so Rdio seems closer to voice-navigability than lots of apps, also a priority (see my answer to What elements of the iOS 7 redesign may reflect future Apple product plans?).
(The Spotify apps, by contrast, are UI-first: constrained window spaces with big buttons and borders, and a tiny little thumbnail of the album art overwhelmed by a garish color scheme that dominates the visual tone).
But above all: it's hard not to look at Rdio and think that it looks like iOS 7. Indeed, it's one of the only apps that seems pretty much at home already on the new OS, without an update: