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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 β to exceed the Luck bound ν ≥ 1/(1−β) for the stability of the MIPT, where ν = 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 β. 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 ψ ≅ β, finding excellent agreement with the conclusions of real-space renormalization group calculations.