I'm taking an extremely basic course in mathematical logic where we briefly talked about nonstandard models of arithmetic. The only example we worked through has been the construction of $PA^\omega$ through the compactness theorem. EDIT: The axioms of $PA^\omega$ consist of the axioms of $PA$ along an infinite set of axioms of the form $\omega > n$ for each standard natural number $n$ (the notation is kind of unfortunate, perhaps I should have chosen a better name than $\omega$ for this symbol...).
I was wondering can this process of "extending" $PA$ to $PA^\omega$ be generalized?
I thought of defining something like $(PA^\omega)^{\omega'}$ adding more axioms of the form $\omega ' > n$ for $n$ in $PA^\omega$ but I'm not even sure this "in" would be well-defined. Is there a way to define such a model?