I don't have an account logged in, but I'm sure that's not a factor. You could try logging it out, but you'd be wasting your time. It's likely just a weird setting. Chrome has a feature that sets everything back to default, so you could try to use that.
Also, I'm quite sure Google links your account with your browsing history, so it might be a better idea to not have an account logged in on your browser.
I wouldn't normally go back and forth in someone's thread, but this is relevant to the auto-logoff feature so I figure it could be helpful to not only myself, potentially OP if they're a Chrome user as well. I ended up logging out to give it a try, which made me lose all my bookmarks, but now the "Stay logged in forever" and "Stay logged in for X minutes" both seem to be working. So in conclusion, being signed into a profile/account through the Chrome browser disables these features and will require you to re-login each time you open the browser.