Are there languages such that A ⊂ B ⊂ C ⊂ D ⊂ E over the alphabet {a,b,c} where:
A is not context-free
B is context-free and non-regular
C is regular
D is non regular
E is regular and not {a,b,c}*
First, let us simplify this and take care of E by just not using c in any language and making E the language (a + b)*. Next, let us deal with D by making it the same as E, but with all strings of prime length greater than two removed. We can choose C to be the set of all even-length strings over {a, b}: (aa + ab + ba + bb)*. For a context-free and non-regular language we can choose the set of even-length palindromes over {a, b}: S -> aSa | bSb | e. Finally, we can choose as A the set of even-length palindromes over {a, b} which begin with a prime number of as.
We might have tried getting rid of D by making it the union of C and some language involving only b, then making C equal to a* and then trying to find A and B using only a... but we might have had trouble finding a context-free non-regular language involving only one symbol.