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As I was doing a few searches today on my forum I realized it may be helpful if the search box remembered prior search terms/phrases that I could choose from.
I think sometimes a browsers auto-complete will try to do this but I know what you're asking and I like the idea.
My reason for reply is if this did happen and it saved them then it should either auto-purge them after a certain time period or a user option to clear their 'saved searches' and the like. Just thinking here as if it appeared without said 'clear' option some may complain.