I've always found the Amazon approach interesting. As part of their sales funnel they email related content to you based on your browsing habits when logged in. Clearly they therefore have a large incentive to persist the user's login status.

Their logout link takes the form "Hello %NAME%. We have recommendations for you. (Not %NAME%?)"

'Not %NAME%' is the logout link, which confuses the hell out of users (just Google 'Amazon logout'). Even if you know it's there, you battle subconscious friction in clicking that link when it is in fact your name.

This strategy has been around years and no doubt adds visibly to their bottom line, even while it exposes users to potential fraud issues with others using their account for one-click purchasing.