The Google results were atrocious for this specific issue, so I decided to create a Q&A here.
I have JavaScript disabled for all websites, and enabled through a whitelist of exceptions.
What do I add to allow Chrome apps/extensions to work?
chrome-extension:// doesn't seem to work, but I'm not sure.
Found through this comment, you apparently add this to your whitelist (for Mac):
file:///<user home directory>/Library/Application%20Support/Google/Chrome/<chrome profile>/Extensions/