The next approach I had looked at was a Hyper-Real type icon which conceptually captures the intention of the application it represents. Examples of this include Instagram, a retro camera which is a direct nod to the types of filters users can apply to their photos, Camera+ which is simply a beautiful camera app, and Evernote Food which is front end app to track and store recipes on the Evernote back end. This type of icon has a lot more opportunity to capture the sentiment and essence of the application. Though, because the icon so beautifully detailed, it is not scalable when it comes to traditional branding practices, but for the purpose of simply representing the application (and not a brand) it works very nicely.
FInally the Branded approach. I probably dont even need to list which apps these represent, because their brand is so strong they immediately stand out. For a designer, there is a little less flexibility in the creativity, but there is still a challenge. The icon is a conduit to the brand by in large, which means it still needs to uphold and represent everything the brand has already established. It seems like a tall order, but a well defined brand should have the collateral to create this already. Most of the time though, this simply means popping the logo on the brand colours. The only [large] caveat here is that for a single purpose app, the designer would need to create an actual brand first.
My Icon Design Exploration
With all that being said, for Notorious (a gestural note taking app, which is also evil) I decided to literally explore every possible permutation of an app icon as seen below.