This isn't really a game, but it will help someone find out where their page ranks in Google - Google Rankings
This is a good one if you speak German - Capture The Map is a strategic search game using Google search.
I'm not sure this is a game, but Google Whack - the elusive search for the Google search that provides only one result.
From the man who made Guess-the-Google, there is also Montage-a-Google which I had before, but has now been updated.
Type in three or four words, and Google Talk will finish your thought.
A different version of Google fighting is to do a Google Battle
Here's a fun little game - fight two words against each other in Google and see who wins! Google Fight
Googlism.com will find out what Google thinks about any subject.