Arduino LilyPad Input Voltages Below the Input Voltage in Specs
The data sheet linked in the Lilypad article for the Atmel AVR microcontroller used specifies that the Atmega 168V (low powered Atmega168) cannot run under voltages of 1.8V. If done so, the internal brownout detection (possibly an analogue comparator where Vin &lt; Vbrownout) is turned on and may trigger a reset signal for the duration of the dip in voltage, which would be always in the case of low voltage supply.

You may possibly be able to go under the 2.7V to >1.8V with success, Red LEDs should work at those voltages too. I am unsure why they rated its lowest at 2.7 as there is no diode voltage drop or anything significant at first glance.


  I understand that an input voltage greater than 5.5 V would damage the device,


Yes, normally a voltage regulator is used to regulate any voltage above nominal down to nominal, for example 5 or 3.3V. These regulators require a higher voltage than output to work (often Vout+100mVish for low dropout, Vout+1V or more for other linear.) 

This regulator is left out all together according to the Lilypad's schematic as to allow lower voltage sources such as coin cells to be used, at the tradeoff of low protection from higher voltages.