I know that some web browsers do not save this information and logs you out of any sites that you have been logged onto during your online activity(ies) automatically. Some web browsers do not save the fact that you have already typed this information into a form, and that it is logged in (although as default, it is normally set on), but you should be able to find an option under tools for this.
Another suggestion is that, say you use a form of private browsing, and choose the option to stay logged in continuously - when you re-visit that same site, the option would have had no effect at all, due to the information being prevented from being stored.