We introduce and study a new model consisting of a single classical random walker undergoing continuous monitoring at rate γ on a discrete lattice. Although such a continuous measurement cannot affect physical observables, it has a nontrivial effect on the probability distribution of the random walker. At small γ, we show analytically that the time evolution of the latter can be mapped to the stochastic heat equation. In this limit, the width of the log-probability thus follows a Family-Vicsek scaling law, N^{α}f(t/N^{α/β}), with roughness and growth exponents corresponding to the Kardar-Parisi-Zhang (KPZ) universality class, i.e., α_{KPZ}^{1D}=1/2 and β_{KPZ}^{1D}=1/3, respectively. When γ is increased outside this regime, we find numerically in 1D a crossover from the KPZ class to a new universality class characterized by exponents α_{M}^{1D}≈1 and β_{M}^{1D}≈1.4. In 3D, varying γ beyond a critical value γ_{M}^{c} leads to a phase transition from a smooth phase that we identify as the Edwards-Wilkinson class to a new universality class with α_{M}^{3D}≈1.