Guardian's Crusade and Misadventures of Tron Bonne, both for PS1. Tron Bonne was about a chick who has to pay off her uncle's gambling debts so she decides to go into the smuggling business, or some other similar backstory.
Guardian's Crusade, well...that was by far one of my favourite playstation games. You play a knight who meets a little pink baby monster while out adventuring, and you take it upon yourself to return him to his mom. Sound childish? Yeah, maybe a little. But on the course of your journey, you have a run-in with cannibals, save a town from being taken over by a corrupt mayor, and help to take down a cult that's taking over the world. Not quite as childish as it may seem. Also, the baby monster was frigging cute!