Surface growth driven by non-monomeric deposition has remained largely unexplored. We investigate a model based on the deposition of blobs with a power-law size distribution $P(s)\sim s^{-\tau }$. We find that the critical exponents vary continuously with $\tau$, recovering Kardar–Parisi–Zhang behavior only for $\tau \ge 3$. For $\tau <3$, roughness scaling exhibits strong corrections and scale invariance breaks down. We show that this behavior originates from the emergence of a second dynamical length scale $\zeta$, corresponding to the linear size of the largest cluster, in addition to the usual correlation length $\xi$. The coexistence of these two relevant scales signals the breakdown of the usual Family–Vicsek scaling. These results point to a new phenomenology of surface growth beyond the standard scale-invariant paradigm.