Triple LOG!

First off, let us just clarify the problem a little bit. $log_{2}(log_{3}(log_{2}x))=0$ The only way that a logarithm can equal 0 is if the logarithm is to 1. Using this info, we quickly derive that $log_{3}(log_{2}x)$ must equal 1, and thus $log_{2}x$ must be equal to 3. Since $log_{2}8$ is the only way we can satisfy all conditions, $\boxed{8}$ is our answer.