Here are sites that have students working together in small teams to do some pretty amazing things.
KudosWall has a list of 50 different student competitions, broken out by areas like Art & Music, Physical Activity, STEM, Volunteering, and Writing.
SolidWorks, a 3D software company, maintains a list of competitions with a focus on robots and STEM activities.
NASA sponsors a ever changing set of student competitions. Check out this resource for the latest list.
Toshiba/NSTA Exploravision contests challenges teams of students to design future technologies. Examples have been: a satellite-based early warning earthquake detection system, a body suit that monitors vital signs and delivers medication.
Every year, Google has a competition open to students grades K-12 to submit artwork for the Google home page. Winners (broken out by grade level) will have their artwork appear on the Google home page.
The Odyssey of the Mind teaches students to learn creative problem-solving methods while having fun in the process. Students develop team-building skills by working in groups of as many as seven students per team.
Have your students submit their work to the various competitions, or use these resources as a springboard to create your own classroom or school wide competitions.