Well, I am not sure about the 'Complete your profile screen', because I am not familiar with that screen. Is it possible that your company added that as an additional screen during the registration process? It is entirely possible if you use Single-Sign On (SSO)
If it was a custom page added into the Community application, then 'yes', there is a way to change that (add more questions / fields). It was typically done in Lithium Studio, or as a very minor customization.
If you are talking about the basic profile page and fields that you can fill out (via the 'preferences' page) in the profile page, 'yes, you can absolutely add additional fields'. Again, can be done though the Studio tool.
