1. Snapseed
Free on iOS and Android.
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.
2. Lightroom
iOS and Android, some functions available for free, or $5 per month for full access.
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
Free on iOS and Android.
Photoshop Express has many of the same features you'd find in Lightroom, including exposure, contrast and color editing options, but strips out some of the pro tools and cloud syncing and, crucially, ditches the subscription fee. It's a great tool for tweaking your images to bring out their best, but you'll also find a decent selection of filters and overlay textures, as well as tools for making cool collages from your images.