Grzegorczyk-hierarchy divides primitive recursive functions in distinct classes with respect to their growth-rate. It seems that the higher we go the hierarchy, the more tools we have to define functions with finite image that can't be defined in the lower levels of the hierarchy. I have been trying to define functions with finite image that exist only in "high enough" in the hierarchy, but so far I haven't succeeded.

How would one define a finite image function for every $i$ that is in $\mathcal{E}_i$ but not in the lower levels?