Plagge was born to a Prussian family in Darmstadt, Germany, on 10 July 1897; many of his ancestors had been military doctors. Plagge's father died in 1904, leaving Plagge, his mother, and his older sister. After graduating from Ludwig-Georgs-Gymnasium, a secondary school that focused on the classics, Plagge was drafted into the Imperial German Army. He fought as a lieutenant in World War I on the Western Front, participating in the battles of the Somme, Verdun, and Flanders. Imprisoned in a British POW camp from 1917 to 1920, he caught polio and became disabled in his left leg. Upon his release, Plagge studied chemical engineering at the Technische Universität Darmstadt, graduating in 1924. He had wanted to study medicine but was prevented from the longer study program required due to his family's financial problems. After graduating, he married Anke Madsen, but the couple had to live with his mother due to their finances. Unemployed, he ran a pharmaceutical laboratory from the house.