Character growth may not be essential to good story telling, but it is certainly an important theme. Without it, a plot may seem somewhat pointless. Why should our protagonist do these things if she or he ends up the same?
Well, maybe just to provide us with some excitement and entertainment. Or maybe to provide a vehicle to make points about something else entirely (politics, philosophy, you name it). Not that there is anything wrong with totally escapist writing, however.
Now in my soon-to-be-released fantasy novel, THE CROCODILE’S SON, our central character, Qala the Pirate Queen (retired), has very little, if any, personal growth. Those around her, the secondary characters, do. This is a valid way to go, and happens with some frequency in a series, where the main protagonist is already well-realized. Tarzan did not ‘grow’ after the first couple novels but he went on to have twenty-some further adventures in book form, with varied characters who changed in some way (and, of course, Burroughs also used his novels to drop in more than a few observations on humanity).
We all do grow. Time changes us in some way or another. If nothing else, we become more knowledgeable. People come and go in our lives, births, deaths, arrivals, departures. We love and cease to love. It can be subtle but every character should reflect this fact.
That is how they become real.