You can't do it with Reader. Using Acrobat you can duplicate the page and even have it automatically rename the fields, so they are unique. As mentioned, that requires converting the page to a Template object and then using a script to spawn new copies of it.

If you don't want to mess around with writing the code yourself feel free to check out this (paid-for) tool I've developed that allows you to do it very easily and quickly: Custom-made Adobe Scripts: Acrobat -- Duplicate Pages of a PDF file (including Form Fields)

With Acrobat Standard or Professional there is a type of page called a template. This type of page can be hidden or visible and can be used to create a copy of a page including a page with form fields. Since form fields with that same name share many properties including the value property, forms created, spawned, form a template can have the fields renamed. The template could also overlay an existing page.

A Lesson in Templates for Adobe Acrobat

When using templates, you should plan your work. With JavaScript or calculations in the fields, you will need to adjust the scripts to account for the renamed structure of the field.