Not sure exactly what kind of games you're looking for, or which age group you're aiming at, so I hope some of this is helpful.
Cell-related crosswords and quizzes: http://www.cellsalive.com/quiz.htm
Activities for younger children: http://www.kathimitchell.com/cells.html
Some worksheets that could be used as a competition:
http://www.teach-nology.com/worksheets/science/cell/
I like this one that asks for organelles to be compared to members of staff in a school:
http://www.teach-nology.com/worksheets/science/bio/work1/
Another couple of quizes:
http://www.biologycorner.com/bio1/cellquiz.html
http://www.tvdsb.on.ca/westmin/science/Sbi3a1/cells/cellquiz.htm
Online games:
http://nobelprize.org/educational_games/medicine/cell/game/
http://fas.org/immuneattack/
A Top Trumps style card game you can buy: http://www.centreofthecell.org/centre/?page_id=226