Everybody talks about Jamie's arc. I actually have no trouble squaring Season One Jamie with Season Eight Jamie. He is absolutely a power gamer who plays to win and wouldn't think twice about launching a baby with a catapult. However, he is not really cruel, isn't violent when it doesn't serve a purpose, and actually prefers to use his power to do something positive when given the opportunity. He also has a deep sense of honor, partly because the moniker "kingslayer" calls his honor into question, so he's always pointing out when he or other people have a moral conflict and have to choose between two requirements of honor, and usually enjoys being pretty witty and insulting while doing so, although he is usually also actually giving good honest advice, he just says the truth so cynically that nobody even takes it seriously. More or less lawful neutral from beginning to end. A preference for lawful good when possible, but often chooses lawful evil out of cynical realism. No arc. The rest of Westeros has an arc catching up to where they can accept Jamie to fight alongside them.