The extremes of space weather are interesting both because they can have severe and unanticipated consequences on our space environment and the technological assets deployed within it, and because such extremes provide us with valuable information about the physical processes involved in the coupled Earth-Sun system in general. The Extreme Space Weather session will focus on solar, interplanetary, planetary, astrophysical, magnetospheric, ionospheric and atmospheric phenomena that we can directly experience, simulate in advanced models, infer from paleo-data (such as ice cores, rocks, and meteorites) and observe from a multitude of stars like the Sun. The extremes of interest include short-term ones, like major space weather events of the magnitude of the 1859 Carrington event, as well as events such as the recent deep extended solar minimum. What are the impacts of such phenomena from the Sun through the heliosphere and then throughout geospace? What are the worst events we should prepare for in terms of geomagnetic storms, energetic-particle populations, or irradiance changes? Should we care about experiencing a modern-day Maunder Minimum? Predictions for the next solar cycle are welcome.