A few months ago I covered Kaomoji, an app that makes it easy to browse and copy Japanese emoticons (also known as, indeed, kaomoji) on iOS. This morning, thanks to Sean's recommendation, I bought Japanese Emoticons, the official app by website japaneseemoticons.net that, like Kaomoji, lets you browse through over 1000 built-in emoticons organized by type (such as "positive emotions", actions, evil, "hello and goodbye", and more).
Unlike Kaomoji, the Japanese Emoticons app lets you add your most-used emoticons to a set of Favorites, which you can rearrange at any time and access anywhere in the app from a button in the upper right corner. Japanese Emoticons also has an editor to create your own emoticon based on parts like eyes, mouths, and arms (there's also a button to create a random one). The app features one-tap copy, but the interface isn't as polished as Kaomoji.
If you've been looking for a Kaomoji-like app with favorites and an editor feature, Japanese Emoticons is $0.99 and Universal.