The bathroom outlet is marked for shavers only - and there are one to two outlets on the vanity/desk in the room depending on the ship. We do bring a power strip so that I do not have to keep unplugging and plugging. Before we started bringing a power strip, there were many times that the camera batteries were not charged due to forgetting to re-plug the charger after getting ready for dinner.
If you have thick hair, you might also want to bring your own hairdryer. The one in the bathroom is not very powerful. If you are in a suite, I believe there is a more powerful hairdryer.