MachineMachine /stream

Types of Games: I have come across various attempts to catalogue the possible varieties of stories that exist. The basic assumption being that all plots (narratives?) can be be boiled down to one among only a few types e.g. Hero leaves the kingdom; Hero steals fire, etc. I am looking for a similar classification system for GAMES, that is: how many fundamental types/kinds of game are there? Note: when I say 'game' I mean everything from checkers, through hide n' seek, pool and soccer up to and including Tower Defence, Super Mario or Halo. I know this might be too broad, but I don't want to limit my enquiry at this stage.

I know a bit about game-theory, but what I want is more taxonomical than this. A taxonomy perhaps of possible game structures / directions / goals / rule organisations that can be applied across game types.

Has there been such an attempt at categorising games? Is it even possible, or should I be thinking about the question in an entirely different way?

Thanks in advance