It really depends on the type of story and like Flying Arrow said, the quality of the writer. If the game has a large cast, I'm willing to accept a few characters fall through the cracks and never get any decent screentime but if its a small cast of six or seven characters and you pull the same B.S. its not as kosher.

As for how much growth, I feel its not so much the quantity as much as its the quality of the growth unless circumstances in the story dictate otherwise at which point, those better be good explanations. This all only works if the player finds it believable, I feel its possible for a character to have significant growth and even though circumstantial evidence does point towards said growth to be logical, if the whole scenario is incapable of maintaining suspension of disbelief then its still bad character growth.

An example would be a cowardly boy who slays a dragon and after the incident becomes very confident and stops whining and being cowardly completely. The character has been through a great ordeal that logically would affect him and raise his confidence but the speed of which his insecurities vanish after said incident and the near 180 degree change so quickly is a bit hard to swallow. Realistically it wouldn't be so neat and tidy. Perfect for a children's story but that kind of crap wouldn't fly in stories targeted towards an older audience.

I feel good character growth is not just how the events affect the character but also how the audience sees the characters. An example is Setzer from VI, who is for most of the game a care-free rogue who seems like he's doing everything just for the fun, then you see his past and your perception of the character is changed cause now his previous actions no longer seem like him being a goofball, but rather a tragic figure who is desperately trying to distract himself from his grief. In truth, Setzer's personal growth is minimal in VI, he eventually comes to terms with Daryl's death but his significant growth is more about how the player sees him, cause after the revelation of his past, a second playthrough of the game changes the context of the figure through hindsight. Still, I feel I should be quick to point out that you need both, not just one over the other.

If the player feels one character is a putz in the game and then they change due to incidents in the plot but it never alters the players perception of them being a putz then the character growth is still bad cause you've just made an unlikable or boring character. If the perception of the character changes due to revelations but the character themselves never deviate from their default personality quirk and even said revelations doesn't add a new level of context to the characters actions then its not good character growth, you've just created a Freudian excuse for them.

You need a little of column A and B, but where I feel many writers drop the bomb (myself included) is that the audience usually never needs dramatic moments to garner such a change, you can do simple things to change a character significantly without all the high emotional drama to garner it in. Red XIII takes a simply journey and is profoundly affected by the revelation of his father but this scene is insignificant to VII's entire story but still it does a great job to help the player grow to like the furball.

High drama does not equal good character growth. It depends on many more factors and the skill of the writer and the audiences willingness to accept the writer's skills to create anything meaningful.