A friend was using Geary with a gmail account a while ago, stopped using it and then started using it again recently. They entered their password and the password dialog popped up again, they tried again and same thing so they hit cancel and logged into the gmail web interface. The first email (attached) was from Google saying that a sign-in attempt was blocked. I got them to repeat the process with network debug turned on (anonymised log attached) and it appears that Google may request users to log in via the web interface at times but Geary doesn't pass that information on to users. This problem happens with both 0.8.2 and 0.8.3.
Woops, actually attached now.
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