Old English growan (of plants) "to flourish, increase, develop, get bigger" (class VII strong verb; past tense greow, past participle growen), from Proto-Germanic *gro- (source also of Old Norse groa "to grow" (of vegetation), Old Frisian groia, Dutch groeien, Old High German gruoen), from PIE root *ghre- "to grow, become green" (see grass). Applied in Middle English to human beings (c. 1300) and animals (early 15c.) and their parts, supplanting Old English weaxan (see wax (v.)) in the general sense of "to increase." Transitive sense "cause to grow" is from 1774. To grow on "gain in the estimation of" is from 1712.
Have you ever heard anything about God, Topsy? ... Do you know who made you?" "Nobody, as I knows on," said the child. ... "I spect I grow'd. Don't think nobody never made me." [Harriet B. Stowe, "Uncle Tom's Cabin," 1851]