Why does this lithium battery have such a flat discharge curve?
The things other people are saying in their answers are true. The regulator is likely dropping out and may eventually cut off, etc. But I think there is one more piece to the puzzle.
I have seen this before. The ADC data are wrong. Once the regulator drops out, VCC is no longer 3.3V. VREF for the ADC is also no longer 3.3V. So all the voltages you calculate after about 20 hours are not the true voltage. What happens is, basically the ADC returns a fixed result for any VBAT lower than around 3.62 volts or so (looking at your graph).
A 12 bit ADC returns Vin/Vref * 4095. If you have a divider on VBAT, then Vin = VBAT * divide_factor.
And Vref = VBAT-VDO where VDO is the dropout voltage.
So the ADC returns 4095 *  VBAT * divide_factor / (VBAT - VDO).
VDO is approximately constant, but since it is small this result changes very little with VBAT.
To confirm, hook the unit up to a lab power supply and take some data at different voltages.
All the data after about 20 hours are incorrect because VREF is no longer 3.3V.