  We investigate iterating the construction of $C(\mathtt{aa})$, the $L$-like
inner model constructed using the so-called "stationary-logic". We show that it
is possible to force over generic extensions of $L$ to obtain a model of
$V=C(\mathtt{aa})$, and to obtain models in which the sequence of iterated
$C(\mathtt{aa})$s is decreasing of arbitrarily large order types. For this we
prove distributivity and stationary-set preservation properties for countable
iterations of club-shooting forcings using mutually stationary sets, and
introduce the notion of mutually fat sets which yields better distributivity
results even for uncountable iterations.
