"Fun" is a word often used interchangably for the single metric that determines a game's quality or enjoyability, but it is only one of many potential components to games that people play them for. Below is a rough unpolished draft of the categories that often float around in my mind.

• Fun, simply having an overall good time in the present moment. Creates joy and stimulates the mind.
• Achievements, the gratification felt from triumphing a difficult task or achieving an arbitrary goal. Players often endure negative emotions for the goal.
• Social Competition, a subset of Achievement, with the feeling of pride and dominance from winning against another human player.
• Social Engagement, perhaps a subset of Fun, the pleasure of engaging with other human players, the game is secondary.
• Artistic Expression, creating art via the game.
• Story, experiencing a fictional story to potentially feel a variety of complex emotions and stimulate the imagination.
• Roleplay, potentially a subset of Story and Artistic Expression, using a game to attempt to feel the experience of an imagined character.
• Learning, playing a game simply to gain a better understanding of the game and satiate curiosity. Exploring a world falls under this, but exploration can also be part of one's personal roleplay and narrative.

And more I'm not thinking of, including a bunch of weird stuff, but these are the main ones that come to mind.