Outlook 2016 loads up with a Google Gmail or Googlemail account configured and imediately prompts with a Sign in with your Google account prompt. It happens as soon as the Send/Receive runs to download new messages.
You might see error 0x800CCC0E Outlook cannot synchonize subscribed folders.
If you login, it prompts to authorise Microsoft, which you can do. It just doesn't seem to retain that setting and keeps prompting. Everything looks ok and correctly authorised if you look at https://myaccount.google.com/security
The account is an auto setup from the Outlook 2016 email wizard when you Add Account, which configures IMAP with servers imap.gmail.com and outgoing as smtp.gmail.com
In fact the same setup worked perfectly on another computer. Recreating the profile, the account, checking OST, etc would not fix the problem.
In the end, the fix was found to be related to the account having previously been a @googlemail.com account. If you manually change the servers to be:
* Incoming server: imap.googlemail.com
* Outgoing server: smtp.googlemail.com
the problem is fixed and your Gmail in Outlook will work normally.