To paraphrase what Frob said, the purpose(s) of a game is whatever you want it to be. Of course, whether your game fulfills those purposes is another question.
People don't really ask about the purpose of books, poetry or film. If they do, they probably are questioning the meaning of life itself. But the fact that people regularly ask this question about games points to deep underlying problems (or features?) within the industry. It points to the massive deficiencies in how game design and development are taught and talked about including the lack of critical analysis of games throughout the industry (including journalism). In fact, when it comes to criticizing certain games or types of games, we see massive push back from folks both within and outside the industry(eg. gamers, media, bloggers). There are lots of people out there that don't want people to look at games with a critical eye. Perhaps because if you dig too deep or peer behind the curtain you'll find Cthulhu (existential crisis).
We know there are all sorts of different types of games. Games can draw from so many different artistic disciplines.
Besides technical design, mechanics design, and art design, games often need sound and music design, narrative (both practical and beyond) and story telling. Games can ask the player to use tactics, strategize, manage resources, perform puzzle solving, exploration, simulation (games can strive to simulate all sorts of things). Each of these areas are worthy of deep study and have been written about extensively.
So to ask such a basic question re: imbuing purpose, shows just how informal game design is. Are most game designers really studying, learning and putting in the work to hone their craft? If I had to guess, I'd say most designers enter their field without adequate study or training. Historically this has been true because in the early days there was little in the way of formal training to be had. I think this is why many of the most critically acclaimed game designers of the past can best be regarded as being highly intuitive rather than well trained. Does that make most game designers hacks? Hmm... maybe. But hacks can make a lot of money too so there's that.
There are a lot of producers and designers who first and foremost, want to make games that are "fun" and dare I say, "addictive." I don't think it's unfair to say that their primary motivation is making as much money as possible. For these people, I think games are more product than art. Today, making games "fun" and addictive is a science that draws upon our understandings of human psychology and neurology. To me, the difference between manipulating someone and educating them is a matter of integrity. Does the teacher (politician, crazy billionaire, news anchor, etc) actually believe the things they want their audience to believe?
Quentin Tarantino is a skilled writer and director and his films tend to be very well constructed from a technical point of view. The characters and stories are well developed. The dialogue is engaging and often provocative and he makes good use of literary devices like foreshadowing and red herrings, but the themes and subject matter of his stories are very...umm.. primordial. But perhaps this is what makes his films so attractive to mainstream audiences.
Taylor Swift has made more money than 99.9% of musicians throughout history so if the $$$ metric is who you pray to, then that is your choice.
And I think its fair to say this is what most game publishers want... they want the technical excellence but they don't really care about the messages and vibes they are putting out there. But, communicating a message(s) can be a significant purpose of a game and aspiring game designers\developers need to know that messages are something that can lead and inform design rather than merely be tacked on after thoughts.
So a game can have multiple purposes including "meta-purposes." Whether you want to enter the Michael Bay school, the Tarantino school, the Kubrick school, some mixture or something entirely new... that's up to you.
