Well, the definition of “entertain” breaks down into two definitions. One being to provide amusement or enjoyment. For most people, that might mean “fun”, but I think it’s easy to see where many people can enjoy a wide range of experiences, even downbeat experiences, that would not be well categorized as “fun”. The second definition is to give attention or consideration to an idea, suggestion, or feeling. This seems much broader, but for me, most of what fits the second definition can easily fit the first. If an idea, etc., is worthy of grabbing my attention, it also provides me with enjoyment, even if the idea, etc., is not something people would label as “fun”.
In fiction, I am often entertained by stories that involve me in perspectives that I could not ordinarily encounter all by my lonesome. Some stories involve identifying with a hero that does awesome stuff and who is rewarded for being awesome. That’s fun. Most stories that I love best, however, explore any of a great variety of situations, many of which are dark and depressing - these stories, while not being fun, nevertheless capture my attention and imagination.
Oh, so many. Unfortunately I have memory deficits, and have a great deal of trouble drawing forth specifics. Probably nearly any Hugo or Nebula award winning story would qualify (and probably many, many others, but I can vouch for good science fiction). I’ve been re-reading the collected works of Theodore Sturgeon, and many of his stories certainly qualify, in that the stories are often very entertaining tales of wonder and mystery, as the reader unravels what is happening in the story, and also meaningful tales that explore an aspect of humanity that is often ignored by larger society. Many of Sturgeon’s “messages” have addressed racism, sexism, homosexuality and even incest. Yet each of these stories are entertaining mysteries or adventures in their own right as well.
It would be easier (in the sense of reducing numbers) to talk about stories (that we’ve actually heard of, at least), where the author valued the message over entertainment and therefore failed to be very entertaining to many people (Ayn Rand’s tales come to mind, as do most “Christian” stories and movies). It would be even easier to talk about those very few stories that value the message overall, but somehow manage to be entertaining, like Animal Farm and, uhm, I’m sure there’s one other.
In the video game realm, I am particularly fond of the Mass Effect series. It’s clearly a story-driven, yet gameplay-aware series which a lot of people find to be fun. Yet there are some clear messages embedded throughout the games, as well. Now those messages might not be anything especially new to most thoughtful 'Dopers, but they also have not yet been explored much in the video game medium. Questions regarding racism (or rather, speciesism), sexuality, and liberty are addressed in the games. Also present are issues regarding allying with morally ambiguous agencies, and, most importantly, taking a step back from one’s righteous quest to reconsider whether an enemy “race” or group could be better viewed as an ally (i.e. the Geth, the Batarians, or any of the Council Races).
And Mass Effect barely scratches the surface regarding all of these questions. And there are certainly more questions to be addressed. I can imagine a great variety of video games that draw from a huge variety of human experiences and imagination that would also be immensely entertaining and popular, and which also happen to bring an enlightened awareness to the consumer about any number of important perspectives.