Despite checking "stay logged in" or the like on Gmail or Docs, Chrome refuses to do so when I close and re-open it with Google sites pinned. If they're not pinned, it works fine.

The "Clear cookies and other site and plug-in data when I close my browser" checkbox in the settings is not checked, and I don't have any cookie exceptions. All settings are defaults. Nor is the incognito mode being used. This occurs on all my computers using Chrome.

I have deleted my cookies file (%userprofile%\AppData\Local\Google\Chrome\User Data\Default\Cookies) with no effect (other than losing the logins that ordinarly work fine).

Of note is that when I relaunch Chrome with Gmail pinned and it asks me to log in, doing so once will fail (does nothing; no errors), then it will work on the second attempt. If I refresh the window before doing so, it will work on the first attempt.