Google-owned Snapseed offers a wide range of exposure and color tools to make tweaks to your images, but also has plenty of filter options, from vintage styles to modern, punchy HDR looks. You can layer the effects up to create some interesting edits on your image. And best of all, it's totally free.
Adobe Lightroom remains an industry standard for professional photographers and the mobile version is much the same. You'll find no stickers, animations or emoji here, but you will get fine grain control over your image and the same set of tools you'd find in Lightroom on desktop. It's the app I use the most to edit my own images on my iPhone and iPad, not least because the images sync in the cloud, letting me start on one device, and continue on another.
3. Adobe Photoshop Express
It's not as open to wild creativity as other options on this list, but it's a solid editing app at a price that's hard to argue with.
VSCO began life making color grading presets for Lightroom and its roots are clear in the app today. Rather than offer stickers and animated GIFs for Snapchat enthusiasts, VSCO is all about the more artful filmic color filters. The app has a huge range of presets available, including looks designed to emulate classic rolls of film from Fujifilm, Kodak and Ilford.
PicsArt has a huge range of editing tools available to you, from basic adjustments like exposure and contrast, through to cinematic color grading and dramatic filters that transform your images into painting-like pieces of art. There are loads of options for both the tone and shape of your face in selfies -- I won't go into the ethics of using these tools for "beauty" purposes, but I had fun in using the tools to intentionally transform my features into bizarre proportions.
There's a whole Instagram-style social sharing element to PicsArt as well, if you're interested in that. Personally I was mostly interested in the editing options.