Personally, I recommend the MDA (Mechanics, Dynamics, Aesthetics) approach and looking at the 8 Aesthetics( Main Reasons why people play games):
1. SensationGame as sense-pleasure (Hearing/music - Catherine, Sight/art - El Shaddai )2. Fantasyas make-believe (Soldier - Call of Duty, Spy - Splinter Cell, Mage/Warrior/Rogue - Skyrim, etc, etc)3. NarrativeGame as drama (For the story, Most J-RPG's, Spec Ops: The Line, etc)4. ChallengeGame as obstacle course (Super Meat Boy, Demons Souls, Any game that's hard)5. FellowshipGame as social framework (Teamwork with other people. Horde on Gears of War, Co-op mode on any game, etc)6. DiscoveryGame as uncharted territory (Wandering around Skyrim, Fallout 3, Sandbox games)7. ExpressionGame as self-discovery (Allow's a person to express themselves. Through choices, in-game creation - minecraft, music creation in Mabinogi, etc)8. SubmissionGame as pastime (To lose oneself in, mindless past-time. Pokemon, Final Fantasy, etc)Personally, I have one more:9. CompetitionGame as competition (Multiplayer in Call of Duty)These are the reasons people play games.Most games have at least three aesthetics. Although some games have fragments of other aesthetics, Mario having story to rescue princess, nobody plays Mario for the story, thus Narrative isn't an aesthetic in Mario.
I agree with you.. and want to add some more games
Pong shot Reloaded on BB, play multi player game with BBM friends.
10. To complete Game missions and Achievements or to complete goals:
There are many games having specific goals, missions and achievements. Player plays to complete those like Jet pack Joyride, Temple Run etc.