The login screen for StackExchange/AskUbuntu has no option to select or deselect a "keep me logged in" function, but it retains the login by default.
Retaining an everlasting login session without the user's permission has got to be some kind of bug or design flaw.
The security implications are obviously less than for a website with financial and/or personal details, but they are still there, and should be locked down.