It's a known issue.
You must bind the items to be able to use more than 5.
ListPicker as an ItemsControl, gets its Items property set to a list of ListPickerItems in your example. ListPickerItems are UIElements, and the ListPicker renders them in its presenter. When there are 5 or less items, the expanded mode opens on the current page, and you can see all the items in the presenter.
But when 6 or more items are present, opening the list picker goes to full mode which opens a new page. This new page has a listbox, which gets its items property set to the listpicker's items. This is where it breaks. By setting the listbox's items to the listpicker's items (in this case a list of listpickeritems), the listbox will put those UIElements into its view. Now a single listboxitem is included in two places on the visual tree.
Because of this issue, ListPicker only supports databinding and templating. DO NOT set the ListPicker's items to specific UIElements.