We study the multitime distribution in a discrete polynuclear growth model or, equivalently, in directed last-passage percolation with geometric weights. A formula for the joint multitime distribution function is derived in the discrete setting. It takes the form of a multiple contour integral of a block Fredholm determinant. The asymptotic multitime distribution is then computed by taking the appropriate KPZ-scaling limit of this formula. This distribution is expected to be universal for models in the Kardar-Parisi-Zhang universality class.