Oh on the computer. Sorry, I thought you were talking about Windows Phone.
Yeah I've noticed that since a very long time ago. Back then it had like an option for save Windows Live ID, and one for save Windows Live ID AND Password, and even when I clicked on that, the next day or whatever it would ask me for my password again.
Most recently though, my Hotmail stays logged in, though Windows Phone asks me to verify my password whenever I want to download an app. Doesn't bother me though, I use LastPass to manage my passwords. It'll autofill the fields for me.
And if it matters, I'm using Google Chrome on Mac.