How can I create a list of 10 items and ask that people select and rate their top five, 1 through 5?

I would recommend you to take a look on the following widgets, I think you may find one that works as you need: 

https://widgets.jotform.com/widget/orderable_list 

https://widgets.jotform.com/app/visual_multi_select

https://widgets.jotform.com/app/dual_listbox 

This guide will help you adding the widget to your form: https://www.jotform.com/help/252-How-to-Add-a-Widget-to-your-Form 

If you have any question, let us know. 

These do seem to provide some of the functionality I'm looking for, but not all of it.  Some of these seem to allow you to select from a list, but not reorder them.  The other one allows you to move them to put them in the order you prefer, but doesn't allow you to select only 5 of the 10 initially.

We understand that you are looking for a way to have your respondents select 5 items out of 10 available items and then have them rearrange the selected items after in the order of their preference. Aside from the widgets, my colleague has provided above, we do not have a field or a widget that allows users to select from a list to rearrange it after. Two of the widgets that were mentioned above, however, allows users to arrange the items as they select the items from the list.