If a game aspires to be more than a simple adventure, if it seeks to have a meaningful story and not just a series of exciting episodes, then it must include character growth of some kind.
The way in which character growth takes place varies by genre. Action games typically restrict growth to new moves and new powerups; the character's mental state does not change. Adventure games, which depend on strong characters and plots, allow for a more literary type of change: personal and emotional growth, unrelated to gameplay. Role-playing games focus on character growth as one of the game's top-level challenges. Role-playing games offer several dimensions for growth:
personal, if the story is rich enough; skills, such as the ability to use magic or weapons; and strength, intelligence, or any number of such character attributes.
To build character growth into your game, you'll have to decide which characters will grow (most often the hero, if there is one) and how they will grow. Physically? Intellectually? Morally? Emotionally? Games use physical growth, in abilities and powers, more than any other kind of growth because it is easy to implement and show to the player.
Then ask yourself how you will implement this growth within the game—through changes in numeric or symbolic attributes, or through changes in the plot of the story, or some other means? How will growth affect the gameplay, if at all? Finally, how will it be represented to the player? Some of your options include displaying numbers on the screen to show the growth (the crudest method), changing the character's appearance, changing the actions available to the player if the character is an avatar, and showing that the character has matured by changing his language and behavior (a more subtle method).