This paper studies a two-dimensional random growth model related to the one-dimensional totally asymmetric exclusion process. The model is shown to have shape fluctuations that, when appropriately scaled, converge in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE). The model can be interpreted as a randomly growing Young diagram, a zero-temperature directed polymer in a random environment, or a first-passage site percolation model. The paper provides explicit formulas for the mean and large deviation properties of the growth process, and proves that the fluctuations of the rescaled random variable follow the Tracy-Widom distribution. This result is compared with similar findings in the context of longest increasing subsequences in random permutations. The proofs rely on properties of orthogonal polynomials and Fredholm determinants, and involve detailed analysis of the Meixner kernel and Airy kernel.This paper studies a two-dimensional random growth model related to the one-dimensional totally asymmetric exclusion process. The model is shown to have shape fluctuations that, when appropriately scaled, converge in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE). The model can be interpreted as a randomly growing Young diagram, a zero-temperature directed polymer in a random environment, or a first-passage site percolation model. The paper provides explicit formulas for the mean and large deviation properties of the growth process, and proves that the fluctuations of the rescaled random variable follow the Tracy-Widom distribution. This result is compared with similar findings in the context of longest increasing subsequences in random permutations. The proofs rely on properties of orthogonal polynomials and Fredholm determinants, and involve detailed analysis of the Meixner kernel and Airy kernel.