Mario Party 6 has certain mini-games and game boards that are different in day and night.
Cozy Grove is synced to real world time, and provides 30-60 minutes of new quest content each day. Games like Pokemon have creatures that change form based on the season on your DS clock. Neo: The World Ends With You for example, offers certain types of pins (upgrades) that level up the longer the game has been off. Other games use the real clock to reward you for not playing. Either way, your in-game play is linked to the real world by the hands of time. Maybe you have to wait for a certain number of hours, or maybe you need to come back at a certain time of day. This can be games like The Longing or Animal Crossing that tie events in the game with the passing of real world time. One way they do this is to make things happen in the game in accordance with the time (and sometimes date) in the real world.
Some, however, want to become a part of our day.
These are often enjoyable because they are an escape from everyday life. Video games create fantasy space in which we play.