C.C.
Oh my goodness, yes. My younger daughter slept like a little angel until about 6 months, at which point she woke up, we realized she had brown eyes, and she started buzzing around like a tiny Tazmanian devil. She has never stopped since and she is now almost 6! We started calling her "the busy bee" because she would buzz from thing to thing, climbing, jumping, rolling, you name it, she was on the MOVE! At 13 months, she climbed one of the bar stools, climbed up on top of the island/bar in the kitchen, and just as I walked into the room, she leaped from the bar top to the back of the couch, which was about 5 feet away. I almost had a heart attack! Another of her favorite tricks was to disable the baby-proof locks on the lower bathroom cabinets, then pull herself up so she had her feet on the tops of the cabinet doors, and then would stand there making them wobble back and forth, giggling at herself in the mirror. It is honestly a miracle that this child has lived this long, AND has never been to the emergency room!
I will tell you, though, that she was a truly terrible, terrible 2! She gave me a run for my money and then some - I believe because she's super smart and her language skills didn't develop as fast as her fine and gross motor skills. She was just super frustrated!
All that said, she did skip Pre-K and went straight to Kinder. Now she is graduating from 1st Grade at the top of her class, and at not quite 6 years old is reading chapter books, can add multiple numbers in her head, etc.
Just keep your little girl as busy as you can. Give her lots of toys that allow her to use her imagination, such as big legos, wooden blocks, heck, almost anything you have around the house will do as well as any expensive toy. Have her help you in the kitchen (give her real tasks to do, like tearing up lettuce leaves, stirring things, etc, as she gets older and more coordinated). Talk to her like she's a grown-up. Make your house as safe as you can for her so she can explore freely. I wouldn't bother with classes, because they will be full of other children who are not as advanced and the activities will be aimed at the lowest common denominator. My daughter was bored silly at Gymboree. Just let her get lots of physical play at home, at the park, etc.
Good luck! It will be a wild ride for a few years. ;)