Chuck E. Cheese for a 1 year old? Hmmmmm. I think you are missing the point of this celebration. At age one, the party is more for you than it is for your child. There is no chance that your baby will ever remember this birthday, aside from pictures and videos that you show her later in life, so why torment yourself (and frankly anyone else who attends, like grandparents, aunts, uncles, etc.) with a kids' pizza place? You have plenty of time for those parties in a few years, when your child has friends who will celebrate her birthday with her. And besides, your child is way too young to enjoy what Chuck E. Cheese is all about - other kids and games!
For the big 0-1, stick with the basics. There will be plenty of memories built right in because it is her first. Get a great cake (it will be her first bite afterall!), a gift that will last with her forever (we gave both of our daughters Tiffany baby combs engraved with their names and birthdays when they turned one - they're only one, they don't know the difference when it comes to gifts, but this is something they can cherish as they age and even pass down) and gear your party toward the adults who will be coming. My guess is that it is you and your family who will want to celebrate this milestone most, not a bunch of kids, so cater your party accordingly. And don't be afraid to have it at home. As I said, there is plenty of time down the road for big kid venues. Plus at home, you can put your child down for her nap whenever she needs it, then return to visit with your guests for as long as you'd like. At a restaurant, you'll be fighting against the clock and worrying more about avoiding a meltdown than about relaxing and celebrating.
Take this milestone and celebrate it for what it is, and do it in an intimate setting surrounded by the people who love you and your child most in this world. Nothing will ever top that - for you or for your baby.
Hope this helps! Have fun planning and congrats on making it through the first year!