I'm actually surprised because it seems like they only do doodles for holidays and the 150th+ birthdays of people that I've never heard of (which is the point, of course). In any case, you can play the game by clicking on the triangle. You can even pick which Doctor you want to play as! The game itself is fairly basic, with the goal being to grab one of the letters in Google's name and make it back to the TARDIS without getting zapped by Daleks and whatever else. I've only played the first level and it was fun.
Hopefully, this won't be like three years ago when Google put a full version of Pac-Man on their front page. Supposedly,
that cost over $120 million in productivity (though really, I think that's a lot overblown bullshit).