Does an $X$ exist so $X \simeq \Omega^n Y$ for all $n$, and X is not an infinite loop space?

Question

Are there examples of spaces $X$ that have the property $X \simeq \Omega ^n Y$ with $Y$ depending on $n$, for all natural $n$, with $X$ not an infinite loop space?

I have a feeling these things should exist, and I imagine they have been studied.