when does redis become more than just a cache?

We're using Redis mainly to cache user session data and some API responses we hit often. It works pretty well, honestly. But I'm wondering about times when Redis is used for more than just quick-access storage. Are there certain ways to set things up or particular jobs where Redis is more central and acts like permanent storage? Has anyone used Redis for this, and if so, what did you have to think about, and what were the good parts?