Infinite non-regular decompositions of regular languages
Your intuition does not match mine, I think this is possible for every initial (infinite) regular language. My idea is that every infinite language has an (infinite) non-regular subset, such that the complement is also infinite. You then can repeatedly find the next language which together form a non-regular decomposition.

How to prove the basic observation? Just set half of the words apart (for the complement, the set of words is countable, so order them). In the remaining words you choose a subset that has ever greater sizes, not growing linearly but at least say exponential. Such a subset cannot be regular.

How do we obtain a decomposition that way: we might omit a string in each step. This is dealt by taking the shortest remaining string in the next language. That choice does not influence any regularity.