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Draining a lithium flat does shorten its life a bit, but leaving it that way for months will *kill* it. (Draining it flat shortens the life vs. having used the same amount but topping it off several times instead.)
Topping off a lithium battery generally doesn't hurt it and may reduce the number of full charge cycles it goes through; however, I've seen studies (probably from battery university) that charging a lithium in the 95-100% range may shorten its life slightly, and some of the charge control chips don't allow that anyway under certain circumstances.
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