OP you want to hold on to every stage because you are aware your child will grow up and no longer be little.
In the mix is that you were able to have one child and no more due to fertility issues and that might be driving a feeling of overwhelming loss for you , alongside loving your DD as she grows up too.
I'm glad you have reported your title to be changed as whilst all parents experience their children growing out of stages, whether they have one or more,( and ultimately get empty nester feelings when they leave home eventually seemingly all too quick) there are some people who lose their children through serious ill health or accidents; some people's teenagers go off the rails and become estranged ; some people can't have children at all.
What you are feeling is a heightened awareness that this is your only child when you wanted more - and it is so important that you get some support (a peer group or counselling) due to fertility issues for that, ......
Because you won't realise you might be putting that in your daughter. Plenty of only children out there that do well and their parent/s enjoy their childhood and don't regret them growing through each stage. I really feel for you, you are not alone, but I. beg you to get some support in case you accidentally and unintentionally let it slip to your child that you struggled as maybe wanted her to stay little forever: and struggle with her growing up to be more independent and all that fear can entail
No matter how many children any of us have, they are on loan to us for the time we get to parent them and help them through each stage. If we do well they will stay part of our family and have close wonderful healthy relationships with us and others if not smothered (or ignored at the other extreme) by their parents.
God yes we all miss the little cute arms and toddler cutie years but they are their own selves who grow up