You need to add up 3 voltage drops:
- The voltage the LM317 needs to work
- The voltage across the reference (1.25V for the LM317) which is also the voltage across R1.
- The voltage across the 0.8 ohm load.
If you are using a 1 ohm resistor, you are expecting 1.25A, so the 0.8 ohm load will drop 1 volt. The LM317 requires 3V to work. So you need a minimum input voltage of 5.25V at 1.25A.
Note that the LM317 has very poor efficiency in this application. 1.25W is getting to the load, but the input power is 6.5W so the efficiency is < 20%. With a higher than the absolute minimum input voltage it's even worse. The LM317 will need a heat sink to dissipate the several watts safely as well.
If you're making a vaping device, I suspect there are a lot of working circuits out there that will be better.