I'm surprised this hasn't been mentioned yet, but Rostislav Grigorchuk's 1980 paper in which he constructs the Grigorchuk group is just under two pages:
On the Burnside problem on periodic groups, Funkts. Anal. Prilozen. 14, No 1 (1980) 53-54.
At the time, no one realized the full significance of this group, but some of the more remarkable properties are proven in the paper.