I've mentioned the Google Lunar X PRIZE many times, and the Android Challenge and competitions related to Google Earth as well. Here are some recent posts on other competitions that Google has run or supported recently. They're all from the "Offical Google Blog":
It's Jam Time - on the Google Code Jam
Gingerbread architecture for all
Gingerbread Competition results are in - on the SketchUp Gingerbread House Design Competition
Make over your site using the wisdom of the crowd - on the Website Workout contest using Website Optimizer
How to become a Dummies.com author - on the Knol for Dummies.com contest:
Simply go to the official contest page, create your own knol about a subject you know well, and the contest judges will select five finalists who will be eligible to have their knols featured on the Dummies.com site. In addition, the best entry overall will be awarded a $1,000 grand prize.
Building engineers - on a Google-supported FIRST Robotics Team called the Thunderbolts