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Motivated by recent experimental studies in microbiology, we suggest modifying the classic ballistic deposition model of surface growth, where the memory of a deposition at a site induces more depositions at that site or its neighbors. By studying the statistics of surfaces in this model, we obtain three independent critical exponents: the growth exponent β=5/4, the roughening exponent α=2, and the new (size) exponent γ=1/2. The model requires modifying the Family-Vicsek scaling, resulting in the dynamical exponent z=α+γ/β=2. This modified scaling collapses the surface width vs time curves for various lattice sizes. This previously unobserved universality class of surface growth could describe the surface properties of a wide range of natural systems.