I assume you use the word "grow" in the sense of develop and mature rather than growing purely in terms of getting taller! This excellent Arthurian narrative is fascinating precisely because it chooses to show us the full development of the central character of King Arthur from his childhood days right through his rule up to his unfortunate ending. White therefore manages to personalise this figure that is so often shrouded in mystery and wonder as we see Wart's gradual movement from youth and innocence to maturity and wisdom.
Let us just remember how Wart is depicted at the start of White's narrative. He is described as something of a keen and enthusiastic youngster with great empathy and a strong moral basis to his character. However, he is also responsive to the tutelage of Merlyn, his teacher, and learns important lessons about life and nature when Merlyn transforms him into a variety of different animals.
We therefore see a significant development in terms of Wart's maturity if we compare his character as a child to the kind of king that he is presented as being. In particular, the biggest change is the way that he has learnt Merlyn's lesson about the way that kings and rulers should never use might to support and sustain their power. Even though Arthur is a character who is presented as being not perfect, he does show significant progression, development and maturity throughout White's Arthurian narrative.