In noisy settings or when listening to an unfamiliar talker or accent, it may be difficult to recognize spoken language. This difficulty typically results in reductions in speech intelligibility, but may also increase the effort necessary to process the speech. In the current study, we used a dual-task paradigm and pupillometry to assess the cognitive costs associated with processing fully intelligible accented speech, predicting that rapid perceptual adaptation to an accent would result in decreased listening effort over time. Both paradigms revealed greater effort for nonnative- relative to native-accented speech, as well as an overall reduction in listening effort over the course of the experiment, but only the pupillometry experiment revealed greater adaptation to nonnative- relative to native-accented speech. An exploratory analysis of the dual-task data that attempted to minimize practice effects, however, revealed weak evidence for greater adaptation to the nonnative accent. These results suggest that even when speech is fully intelligible, resolving deviations between the acoustic input and stored lexical representations incurs a processing cost, and adaptation may attenuate this cost.