The point that you REALLY, REALLY, REALLY shouldn't go below with a 3.7V lithium is 2.7. That's what PCB based overdischarge cutoff circuits are set to. That's why if you have a battery that measures 3.1V on your meter, and you try to use it in, for example, certain power hungry flashlights, you'll get a quick flash of light before the circuit cuts the battery off. The loaded voltage is 2.7 or less.
2.7V is a bare minimum, and you should never let your battery get that low. Even if it's not dangerous, it significantly reduces the batterys life. Personally, I try not to let it drop below 3.5V with no load. If I'm sitting around the house, I'll swap batteries when I get to 3.7 or even 3.8 or 3.9V. Right now, I can vape for several hours on an 18650 before the voltage drops from 4.2 to 3.9.
That's the good thing about lithiums. Unlike the old Nicads, they have no memory effect. You don't need to deplete them before recharing them to keep them in good shape like the old Nicads. Charging them 10 times when they're 10% depleted only counts as 1 cycle. With Nicads, that would be the same as depleting them and recharging them 10 times, so 10 cycles would be cut off their life, even though you've only used 1X their capacity.
Basically, you cannot recharge lithiums too soon or too many times. A cycle is only counted as the total capacity drawn out and put back in.