Iterating the cofinality-$\omega$ constructible model

We investigate iterating the construction of $C^{*}$, the $L$-like inner model constructed using first order logic augmented with the "cofinality $\omega$" quantifier. We first show that $\left(C^{*}\right)^{C^{*}}=C^{*}\ne L$ is equiconsistent with ZFC, as well as having finite strictly decreasing sequences of iterated $C^{*}$s. We then show that in models of the form $L^{\mu}$ we get infinite decreasing sequences of length $\omega$, and that an inner model with a measurable cardinal is required for that.