When engaged with a specific task, humans and other animals occasionally make seemingly irrational decisions known as "lapses", even in static and well-learned environments. Given they serve no obvious reward-maximizing purpose, these choices are commonly attributed to disengagement, distraction or sensorimotor noise. However, "lapses" could also be related to a decision-makers' own, hidden objectives. We know, for example, that exploratory objectives can better account for lapses than undirected noise in certain models. Critically, however, there are multiple mechanisms for exploration and we still do not know if exploratory lapses are truly selective, information-seeking decisions, or else the result of exploratory noise processes. To tease these hypotheses apart, we used a novel optimal stopping task in which decisions are made serially rather than simultaneously. We found that 2 rhesus macaques sacrifice objective reward in order to generate lapses, even in fully deterministic reward conditions. These results suggest that lapses are indeed subjectively valuable, selective decisions. However, these results also challenge current models of decision making, which do not allow for lapses after learning when there is no ambiguity in reward outcomes. We show that an ideal, reward-maximizing agent would always wait for the best option to appear once it has been identified in this task. To understand lapses, we developed a novel model that optimally trades off maximizing reward rate with information rate. The model can account for the occurrence of lapses, even when monkeys have learned the value of options and when there is no ambiguity in reward outcomes. These results suggest that there is an intrinsic motivation for information, both when it is useful and when it is not, which can extend our understanding of behavior as a product of intrinsic drives.