A game is above all a set of rules dictated by its designer. The most direct link one can have with the designer’s ideas is through the written rules—if they are properly written of course. They are the “pith and marrow” of the game.
Personally, three kinds of games are of particular interest to me: games that create a new genre, games that summarize the best of a given genre, and games that offer a variant simple enough to engage new players.