Why not have every lesson passed unlock a casual game that you can play at any time from within the Duolingo site?
Would this make Duolingo significantly more addictive?
Where would all these games come from?
Some (many?) iOS/Android developers may be willing to allow Duolingo to host online versions of their games as a way to market their games for mobile platforms.
Duolingo could also share its profits with 3rd party game developers.
As long as they are entirely optional and not intrusive, Having spent maybe an hour translating a hard tract to improve my French, the last thing I want to do is spend more time with the computer. Those with really limited attention spans need not apply! To be honest, the Duolingo developers need to spend all their time on improving the UI and scoring algorithms, not on bells and whistles.