When I think of collages, I think of apps that allow me to arrange individual photos into a new composition. Bazaart (free) does something else. You can take individual elements from photos and make a new photo from those parts.
Usually, this is very difficult because removing an element from a background can be difficult. Even Photoshop isn't perfect at this, but it does provide tools that make it easier.
Bazaart is the best I've seen at this task. I took a picture I had of my parrot, tapped on him, and Bazaart figured out how to grab the image, then let me put it in the foreground of an image from the desert. See the slide show below for the result. It was quick, clean and easy, and if you need to manually clean things up you can do that too.
Bazaart photo editor for iOS
The app also allows you to add text, and control the transparency of your picture elements.
You can also just as easily remove a background and replace it with something else, leaving the foreground intact. Since the app has a social element, you can explore the photos of other people.
When you are done, save your image to your camera roll, email it, or send to Instagram, Facebook, Twitter, What's App or Pinterest.
The app has some videos that explain how it works.
Bazaart will stimulate your creativity and make some usually hard photo editing functions quite easy. The app is universal, requires iOS 7 and it is optimized for the iPhone 5. Bazaart offers some additional fonts as an in-app purchase, but there is a nice free selection.