When I first learned about fundamental groups at Canada/USA Mathcamp during the summer after 11th grade, our teacher suggested that we had already known about monodromy for years. He was explaining covering spaces, and he asked if we had done the following when we were 5 years old. Often children will find a themselves in a building that's not simply-connected and then wonder, after going around a loop, whether they're in the same place or an identical copy of where they were before. Or I've experienced games in which children go around a column and then feel that they have to go back around the column the same number of times in the opposite direction in order to be back where they were before.