I suggest you remove all your cookies for geocaching.com to see if that helps, while at it, you can remove those from project-gc.com as well. I doubt the later will help since we don't store anything interesting in them, but due to the same reason, they can just be removed as well.
From our knowledge and experience this is not a common problem. Chrome is one of the two most commonly used web browsers and I believe most only has to authenticate a few times per year (due to different various reasons).
It is however known that some can't login at all until they remove the geocaching.com cookies. From our understanding, this is an issue with geocaching.com and nothing we can do anything about. But, since we have never had the issue ourselves, we haven't been able to actually research it either.