“Grow” is classified as a change-of-state copula but by definition, copulas may be followed by adjective phrases, noun phrases, or adverbial prepositional phrases. “Very quickly” is an intensifier (degree adverb) with an adverb. Does “grow” still fall as a copula or is it a lexical verb?
In a comment, John Lawler wrote:
The meaning of grow that some might call a ‘copula’ is the sense of “growing to become” as in She grew tired. This is not the sense of grow in Babies grow quickly.