Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.

This is an attempted rephrasing of question: Chain/Hierarchy of Monoids. My application domain is reasoning about modifiers of modifiers in software product line engineering, thus lacking established mathematical background. I find it easier to adapt existing results, even if the application domain is significantly different, so any help would be appreciated.