After updating to the newest Sourcetree (2.6.10), when I went to open my existing Github repositories, it requested my Github credentials.  I entered them correctly and then Sourcetree again prompted me for my Github credentials.  This continued in an infinite loop until I gave up on attempting to connect to my Github account.

I was able to stop this from happening by uninstalling Sourcetree, deleting my Atlassian folder in AppData/Local, reinstalling Sourcetree, and then adding my Github account back in.

I tried other various solutions from within Sourcetree like removing my Github account and adding it back in, but it would not work until I deleted the Atlassian folder.  I did not try just deleting the Atlassian folder (without reinstallation), but I have a hunch that would work.

I had the same thing happen on my laptop, desktop, and Surface Pro 4, I teach game development and 3 other students in my class (so far) have had the same issue.  Most of the students did not have an older version of Sourcetree, so I assume that is why they didn't encounter the issue.

This same issue has actually been re-occuring since the initial release of Sourcetree 2.0.  Each time I upgrade Sourcetree I have this same problem and I have been able to work around it the same way for about 3 years.  It would be nice for this issue to actually get fixed.

Feel free to follow up with any questions.