A common obstacle for applying computer vision models inside the vehicle cabin is the dynamic nature of the surrounding environment, as unforeseen situations may occur at any time. Driver monitoring has been widely researched in the context of closed set recognition i.e. under the premise that all categories are known a priori. Such restrictions represent a significant bottleneck in real-life, as the driver observation models are intended to handle the uncertainty of an open world. In this work, we aim to introduce the concept of open sets to the area of driver observation, where methods have been evaluated only on a static set of classes in the past. First, we formulate the problem of open set recognition for driver monitoring, where a model is intended to identify behaviors previously unseen by the classifier and present a novel Open-Drive&Act benchmark. We combine current closed set models with multiple strategies for novelty detection adopted from general action classification in a generic open set driver behavior recognition framework. In addition to conventional approaches, we employ the prominent I3D architecture extended with modules for assessing its uncertainty via Monte-Carlo dropout. Our experiments demonstrate clear benefits of uncertainty-sensitive models, while leveraging the uncertainty of all the output neurons in a voting-like fashion leads to the best recognition results. To create an avenue for future work, we make Open-Drive&Act public at www.github.com/aroitberg/open-set-driver-activity-recognition.