September 2021 note: SourceTree is a clumsy bug-ridden mess that has only marginally been improved in the four years since I made this post. The great news is that you can uninstall and delete SourceTree from your mind because Visual Studio now has all the Git source control functionality you need integrated into it.
Sometime in the last few weeks SourceTree 1.9.x for Windows started popping-up the Authenticate dialog prompt, over and over and over and over... You could you enter your password until hell froze but it just kept asking.
I noticed that Tools > Options > Authentication had an empty list of saved passwords, whereas for the last couple of years I remember seeing my user name in there.
Countless web searches on this problem revealed lots of complaints about it, but all the discussions and suggestions were utterly irrelevant or useless. Going through the tedious steps of installing SSH keys did not help.
Out of frustration and disgust I decided to uninstall and purge SourceTree completely from my PC. I noticed that there were 3 icons with different versions in the Programs and Features list and they did not uninstall cleanly. Then I searched my whole C: drive for anything with the word 'SourceTree' in the name and deleted it. I then ran regedit as Admin and similarly deleted any keys or values containing 'SourceTree'.
After reinstalling the most recent copy of SourceTree and entering my fresh credentials the problem went away and the program now seems to be running trouble free.
SourceTree issues an irritatingly frequent number of incremental updates, which I usually accept, so I'm guessing that an accumulation of these updates has corrupted something. I'm angry that I had to use such a ham-fisted fix for this, but I document it here in case it helps others who suffer this problem.