Currently, an extension I've made can help show off a channel, including showing off their last played game. As this is one of the main reasons a person may be interested in following that streamer (based on the information that can be gathered by the API), the main way to improve upon that is to show more of these games.
It seems strange that there is a list of the user's last several played games freely available on their channel, but not to be retrieved via an API.