I love taking RAW photos with my Olympus PEN and processing them with Capture One Pro. Like many people, however, my ever present pocket camera is my iPhone, and there are countless photo processing and editing apps available. I've settled on a handful of apps that I use, with the goal of being to get great results in a minimal amount of time, while still being in control of the process (no one-button filters for me).
With the exception of Photosynth, I always take photos with the default iOS camera app. It seems to start up faster than other apps, and I love that I can access it from the phone's lock screen.
The two go-to processing apps I use are CameraKit and Big Lens.
I've been using Camera Kit since back when I had an iPhone 3G. The app is still maintained by its developer, and while it isn't the fastest or most fully-featured app, it still has a great vintage film style that is unlike anything else out there.
I really love Big Lens. Its main purpose is to let you add depth-of-field blur to an image, and it does a superb job of it. A number of apps, like Instagram, let you add a pseudo tilt-shift blur, but Big Lens actually simulates a wide aperture. Importantly, it also includes a number of great-looking color correction filters, particularly the Lomo-style filters. Together, the DoP and filters give iPhone pictures a pretty believable DSLR look.
Other Great Apps
I use a few other apps for a variety of special purposes.
Cortex Cam is a great "long exposure" app that let's you take nice-looking low-light photos without grainy high ISO noise. I use it a lot to take photos of wine bottles at night that I upload to Delectable.
I rarely try to create fake HDR, but when I do, I use Dynamic Light. Like most HDR post-processors, it can overdo the effect, but sometimes that can make a boring shot more fun. Dynamic Light also has a number of other filters and effects, but I never use them.
iOS 7 includes a basic panorama photo mode, but Photosynth is a lot more powerful. With more practice and patience, I hope to make some Widelux-style pictures (a la Jeff Bridges). I sometimes like to process exported Photosynth pictures via Dynamic Light.
Snapseed is definitely my favorite general purpose photo editor. Once you get accustomed to the user interface, you can make a lot of power edits and corrections very quickly. I mostly use Snapseed with photos that I import from my Olympus PEN via an Eye-Fi card. It almost seems specialized for that purpose.
I tried using Apple's Photostream for sharing, but found it really difficult to use, and often unreliable. And of course, only Apple users can see the pictures.
So I post my processed iPhone photos to Instagram. I used to cross-post them to Flickr, but I've decided to use Flickr primarily for my Olympus PEN photos or for albums to share with families. Instagram has always had some decent filters, and the latest has some serviceable editing tools, but since I process everything in other apps, I almost never use an Instagram filter. I do wish Instagram supported non-square photos, especially for panoramas.
When I want to send my photos to someplace other than Instagram, such as Dropbox or Google Drive, I upload them via PhotoSync. PhotoSync is the Swiss Army Knife of photo-uploading apps. If you want to send a picture somewhere, PhotoSync can do it.