To have knowledge or skill gained through long experience. “Eyeteeth” are the canines, which lie directly under our eyes. They cut through the gums when we were very young, so we've had them for almost as long as we've been alive. To ask a pianist how long she played ragtime might be answered with “Oh, I cut my eyeteeth on that kind of music” In other words, for a very long time. Like many other childhood possessions, eyeteeth are so prized that “I'd give my eyeteeth” means to exchange a valuable asset for something that's highly desirable.