Getting a compilation error when using a MFC CList with CStringArray elements

The error message you get means that CStringArray does not have an assignment operator (operator=).

CList seems to be attempting to use such an assignment (possibly to copy elements around) and therefore you get a compilation error.

As suggested in comments, a possible solution is to use standard C++ containers instead. This is recommended anyway since MFC's container classes are quite ancient and not necessarily up-to-date with modern C++ features.

Since CList lists behave like doubly-linked lists, you can consider to use std::list for it. And for a "sequential" (array-like) container it is recommeneded to use std::vector (or std::array for fixed sized ones).

#include <list>
#include <vector>

//...

std::list<std::vector<CString>> aListOfStringArrays;

Note that a linked-list is less cache-friendly that an array-like sequential containers and therefore usually less efficient. So you can even consider to use std::vector instead of the outer std::list:

std::vector<std::vector<CString>> anArrayOfStringArrays;

A side note:
If you want to introduce types for the list/array, instead of typedef you can use the more modern C++ type alias, e.g.:

using CListOfStringArrays = std::list<std::vector<CString>>;