Geer was born in Frankfort, Indiana, the son of Katherine (née Aughe), a teacher, and Roy Aaron Ghere, a postal worker. His father left the family when he was 11 years old. He was deeply influenced by his grandfather, who taught him the botanical names of the plants in his native state. Geer started out to become a botanist, studying the subject and obtaining a master's degree at the University of Chicago. While at Chicago, he was a member of Lambda Chi Alpha fraternity.