Approximate computing is an emerging paradigm enabling tradeoffs between accuracy and efficiency. However, a fundamental challenge persists: state-of-the-art techniques lack the ability to enforce runtime guarantees on accuracy. The convention is to 1) employ offline or online accuracy models, or 2) present experimental results that demonstrate empirically low error. Unfortunately, these approaches are still unable to guarantee acceptability of all application outputs at runtime. We offer a solution that revisits concepts from anytime algorithms. Originally explored for real-time decision problems, anytime algorithms have the property of producing results with increasing accuracy over time. We propose the Anytime Automaton, a new computation model that executes applications as a parallel pipeline of anytime approximations. An automaton produces approximate versions of the application output with increasing accuracy, guaranteeing that the final precise version is eventually reached. The automaton can be stopped whenever the output is deemed acceptable; otherwise, it is a simple matter of letting it run longer. We present an in-depth analysis of the model and demonstrate attractive runtime-accuracy profiles on various applications. Our anytime automaton is the first step towards systems where the acceptability of an application's output directly governs the amount of time and energy expended.