Suppose a user wants access to a document that he does not have permission to see.
He fills out a form in SharePoint that requests access. It sends an email to his supervisor, who has the option to either grant or deny access to the document.
You can, but you'll need to surface a list of documents somehow so that the user can select it for approval. This will require custom code as you'll need to use impersenation to fetch the items as the user wouldn't have access to the library in order to request access to it.