  We study one-dimensional hybrid quantum circuits perturbed by quenched
quasiperiodic (QP) modulations across the measurement-induced phase transition
(MIPT). Considering non-Pisot QP structures, characterized by unbounded
fluctuations, allows us to tune the wandering exponent $\beta$ to exceed the
Luck bound $\nu \ge 1/(1-\beta)$ for the stability of the MIPT, where
$\nu=1.28(2)$. Via robust numerical simulations of random Clifford circuits
interleaved with local projective measurements, we find that sufficiently large
QP structural fluctuations destabilize the MIPT and induce a flow to a broad
family of critical dynamical phase transitions of the infinite QP type that is
governed by the wandering exponent, $\beta$. We numerically determine the
associated critical properties, including the correlation length exponent
consistent with saturating the Luck bound, and a universal activated dynamical
scaling with activation exponent $\psi \cong \beta$, finding excellent
agreement with the conclusions of real space renormalization group
calculations.
