How expensive is access to database? How often do we access to it?

Im about to write an application for Android, and it will use Mysql. I know that access to DB is really expensive in terms of time, and would like to know how often do applications like instant messaging, online gaming access to databases? For example in a game, we would like to save the positions of a player in the world, when he's moving all the time.

Is the database access actually not expensive, and there is a way to be connected to it all the time and just do request that are actually not expensive?

Or is IT really expensive in anyway, and there are techniques to access to it for example every X interval of time, and saving it locally in the meantime?

Thank you