Provide an API to retrieve recent user notifications (likes, RTs, follows, added to list, mentions, etc)
Today we have an API to get mentions and a cumbersome API to get some RT info. However, constructing something equivalent to the "Notifications" tab in Twitter is difficult or impossible. Where it's possible to approximate this functionality, it's inefficient (involving many requests) and runs up against rate limits very easily. A simple "GET notifications" API which returned objects representing notifications of all of these types (optionally with some filtering capability perhaps) would benefit app developers and likely reduce load from people bending over backward trying to replicate this functionality through the current APIs.
Previously, it was possible to recreate this experience using the Streaming API combined with a dedicated service (which I had built), however when the Streaming API was deprecated this solution was no longer viable, and my app's functionality suffered greatly.