I don't have an Archive button. perhaps because I chose to remove it. Where does it appear, when it decides to?

It appears in the line with the other buttons such as replay/

choose email that does not have archive button (I'm assuming this email has not already been archived.)

Try this: Right click in area where buttons display and select 'Customise' click on 'Restore default set'. click on 'done'

Thanks for the suggestion but it does not work. When I go to customise is shows the archive button but when I click done the archive button goes. Clicking restore to default makes no difference.

Start Thunderbird in Safe Mode and report back on whether the Archive button is now appearing.

In Thunderbird Help > Restart with addons disabled.