As a rule of thumb, you should never allow Li-Ion batteries like the LP-E6 to drain completely. Lithium Ion cells have improved longevity if you constantly top them up, or in other words re-charge them when they reduce to about 15-20% charge. Allowing the batteries to be used up completely - squeezing that last shot out of them - will reduce the number of complete recharge cycles that you'll get out of them.
I'm curious if Canon makes the camera shut off at a certain voltage, above the trigger point for the protection circuit in the LiIon. Seems like it would be easy and allow you to run the battery until the camera shuts off.
I don't fully buy into this hype about topping off all the time. Yes, read plenty of articles on it, most of them regurgitation of some points made in a article on battery university. I remember some stats on recharge cycles vs percentage of discharge, and of course the 5% or 10% discharge had many more recharge cycles... but nobody points out that you only get to use 10% of your battery. It better have a lot more recharge cycles, at least 10X just to break even, and more to make it worth my while.
I usually pull my batteries out when I get the low battery warning, if I have a spare handy. If I don't I won't hesitate to ride it until it turns off. Maybe if you burn through a battery a day, 5+ days a week I can see how it'll add up. But LiIon batteries start degrading the moment their built, most specs I've seen give them somewhere between a 2 and 5 year lifespan. 1000 recharge cycles, 5 years - 200 a year. Whatever, beats popping them into the charger every night after a small shoot.