The subject of exactly which Character Arcs are possible in a game is one of those vast topics that goes beyond the scope of this book. There are many viable and interesting ones. We've already discussed two:
Here are few more.
To Become a Leader
For instance, your character could start out as a grunt, a no-account, and get rewarded for each act of decisiveness. A leader needs followers, and so you'd acquire followers as you move through your Character Arc.
To Learn You Are Worthy of Love
You start out hated. Your father had been ruthless, but your acts of heroism and goodness are rewarded to the point where you become the most beloved person in the land.
To Achieve a Spiritual Connection to the World
You probably can't try for a more challenging First-Person Character Arc than this. After all, do we dare suppose that you can give a spiritual experience to a player?
Yet it must be possible, for it occurs in films like Crouching Tiger Hidden Dragon and The Karate Kid. Some people might even put the films of the Lord of the Rings trilogy in that camp or the film Phenomenon.
Here are few potential approaches. In all of them, you start out just as you are. You are rewarded, however, for:
The rewards themselves would have to have a spiritual quality to them, such as:
Of course, some might argue that these abilities and rewards might seem interesting (or magical) but not have a spiritual quality at all. It's all in the execution. For example, Crouching Tiger Hidden Dragon could have been completely lacking in spiritual overtones in the hands of a lesser writer and director.
Also, the spiritual overtones in the game wouldn't just come from the experiences that lead to spirituality or in the particular rewards, but in the linking of these two.
To Learn Ethics
You start out as a pirate, out for only yourself, but you receive increasing incentives (rewards and punishments) that guide you to make ethical choices including eventually, perhaps, turning on the pirates you used to travel with and who've hurt or killed many innocents.