If you’re going to define it at all, the only reasonable definition is that $\wp^{(\eta)}(A)=\bigcup_{\xi<\eta}\wp^{(\xi)}(A)$ when $\eta$ is a limit ordinal.
For $n\in\omega$, elements of $\wp^{(n)}(A)$ are (intuitively speaking) sets that have members of $A$ buried inside $n$ layers of curly braces. You can’t have things buried exactly $\omega$ layers deep, because you need an outer layer of braces, but you can have things buried at most $\omega$ layers deep, though of course that just turns out to be things buried less than $\omega$ layers deep.
It’s more natural to look at the operation $A\mapsto A\cup\wp(A)$: then it’s clear that at limit stages you just want to take the union of the earlier stages.