I'm not sure if this will help, but here's a suggestion: Go to ~/Library and see if you have a Google folder. In my system, there is a GoogleSoftwareUpdate folder within. You might try trashing it, if you have it. I also have a Google Earth folder in ~/Library/Application Support. Take a look there as well.
EDIT: I just found a Google folder in /Library, and it also contains GoogleSoftwareUpdate. A good way to find everything Google (including hidden files and folders) is by using
EasyFind.
This seems to have worked, but had to I trash both of those folders. I dumped the one in ~/Library/Application Support last night and this morning there were 4 (yes, FOUR) popup alerts on my desktop. Dumped the second folder in /Library and that seems to have stopped the annoyance. The only strangeness was Chrome alerting me that it could no longer auto-update without my permission. That was easy to fix. Thanks, Jon!!!
Did you attempt to just log in as you would if you were starting up or installing an application? I just had the same pop-up happen and logging in took care of it.
Yes, didn't stop the alert; but Jon's tip did the trick.