Kathryn Schulz, author of Being Wrong: Adventures in the Margin of Error, makes a very eloquent case in her book and her TED talk for the benefits of living in a world "rich with error." It is only, she argues, by exploring and making mistakes that we learn and grow.
I think most of us would agree with this on a purely intellectual, thought experiment, ideal world, level. Unfortunately our emotions are frequently in conflict with the contents of our heads. We are afraid to be publicly wrong. Many of us grew up in worlds where to be chastised for being wrong is to be downgraded to a mere child and to experience humiliation at the hands of people more powerful than we are.
THAT is exactly what we are afraid of when we cross cultures for the first time. And we are right to be a bit cautious because crossing the cultural divide is guaranteed to put you on the "path to wrongness."
Why? Because outside of our native cultures we are all cultural incompetents in the beginning. Coming to Tokyo for the first time you learn fast that a 8-year old child in Japan knows more about swimming in Japanese culture than you do. Furthermore, she can read. You, on the other hand, are now a 40-year old illiterate. This is a bit destabilizing for an adult.
It is also liberating once we get over the shock. The moment you admit that you are incompetent in this context, you can relax and start learning as a child learns - by trial and error with a sense of wonder and curiosity. As Schulz says, "Sometimes we want to be the toddler in Times Square. We travel to feel like a kid again: because we hope to experience the world as new and because we believe the best way to learn about it is to play in it."
And that is where we come to what I think is the very best part of crossing cultures. As you gradually become culturally competent, this new world really does become your playground. Once you are comfortable with another culture, you can start to play with it, to have some fun. Because you come from the outside you can see possibilities and opportunities that the natives don't see because they don't have sufficient distance from their own culture to be flexible and challenge the rules.
Every migrant is an amateur anthropologist and a quiet rebel against his adopted culture(s). We are the people that drive the natives crazy with "Why?' questions. Since we learn about the target culture from the ground up, we are keen observers and imitators. This experience is cumulative - the more cultures we experience, the faster we find our feet. We get very good at being consciously incompetent. We know that we don't know.