Self-triggered control is a promising aperiodic control paradigm that allows the design of resource-efficient control schemes. Several results were recently developed for self-triggered control of linear systems, but self-triggered control for nonlinear systems is still not well understood. Existing techniques are mainly limited to homogeneous or polynomial closed-loop systems that are input-to-state stable with respect to measurement errors. These assumptions obviously limit the class of applicable cases. In this paper, a new simple self-triggered controller, based on the assumption of local asymptotic stability of the closed-loop system, is presented. The controller ensures local ultimate uniform boundedness of the state trajectories in the sense that the self-triggered controller guarantees the trajectories to be confined into an arbitrary small region. In the case with measurement time delays, it is shown how the size of the region depends on the maximum delay. The effectiveness of the proposed method is illustrated by simulations.