Jun3 2016 update: as of right now I only use the A Color Story app! It's 100% amazing and worth the money. It's available for iPhone and soon to be on Android!
For the majority of my iPhone photography, I use three apps:
Camera+ has replaced the normal camera app for me - it has more options, including an image stabilization feature that's currently missing for the default camera app on iPhone 4S/5C. It has loads of options to enhance your photos, but it's also just fine to use it without making any adjustments.
Afterlight is great for tweaking photos! I use it for cropping, adjusting brightness and colors, and also for adding frames. The app also has a fantastic set of filters that are actually useable - they don't make the photos grainy and strange like Instagram does.
Instasize is a great way to post full photos to Instagram and social media without cropping them into a square. So many times I take photos and love the way they look and I don't want to butcher them by cropping - and this is a perfect solution! As an added bonus, you can choose the color of the borders around your photo, but I tend to stick with white. :)
Free (and awesome!) photo editors:
Pixlr and Photoshop Express are capable of creating very pretty and well edited photos. I haven't noticed any issues with pixelation when the images are blown up. :D
Litely is not as full featured, but it's lovely for adding a bit of mood and polish to portraits - it's essentially just a collection of really beautiful film-style filters.
If you want to add text to your photos, I recommend using one of these apps:
The one major drawback here is that most text editing apps default to square cropping and/or can be a little clunky to use.
A Beautiful Mess also has a hard time processing edited photos - I find that when I blow the image up it's quite pixelated. But if you're just using it for Instagram, it might work just fine for you!
Typic+ also has a free version with less options if you want to give it a go before you buy. :)