I'm trying to prove $EXP \subset E^E$ (strictly).
I believe I need to construct my own $A \in E^E$ and show that $A \notin EXP$, but I cannot think of a smart way of doing that.
The class $E^E$ contains 2E, which is the class of problems solvable in iterated exponential time. The reason is that the E oracle can be applied on a padded input of exponential size. The time hierarchy theorem separates 2E from EXP.