A way for user to choose fields?

I couldn't find anywhere else to post this question, so I'm posting here in General.

I want to create a content type where the user can arbitrarily choose which fields (or perhaps field collections) to add in any order. So if there is a field type A and a field type B, the filled out form may look like

A
A
B
B
B
A
B

or whatever the user desires.

I imagine the workflow would use "add another" and then have a dropdown with a choice of which field to add. And perhaps the user could reorder them with drag handles. I've searched for a module that has this functionality and haven't found anything. I would imagine I am not the only person with this problem. Maybe there is a simple solution that I am missing?