How to get Word to display more than the first 3 headings in a Table of Contents

The trick is to hit "Alt F9"  when you are focused on the table of contents.

This will produce something like:

{ TOC \o "1-3" \h }

Change the 3 to a 4 and you're done.

Exit the TOC field by hitting Alt F9 again and rebuild the TOC, you're done