Sporadic solar energetic particle (SEP) events affect the Earth's atmosphere and environment, in particular leading to depletion of the protective ozone layer in the Earth's atmosphere, and pose potential technological and even life hazards. The greatest SEP storm known for the last 11 millennia (the Holocene) occurred in 774-775 AD, serving as a likely worst-case scenario being 40-50 times stronger than any directly observed one. Here we present a systematic analysis of the impact such an extreme event can have on the Earth's atmosphere. Using state-of-the-art cosmic ray cascade and chemistry-climate models, we successfully reproduce the observed variability of cosmogenic isotope 10Be, around 775 AD, in four ice cores from Greenland and Antarctica, thereby validating the models in the assessment of this event. We add to prior conclusions that any nitrate deposition signal from SEP events remains too weak to be detected in ice cores by showing that, even for such an extreme solar storm and sub-annual data resolution, the nitrate deposition signal is indistinguishable from the seasonal cycle. We show that such a severe event is able to perturb the polar stratosphere for at least one year, leading to regional changes in the surface temperature during northern hemisphere winters.