What is the most frugal way to charge a single cell li-ion battery?
A current limited voltage supply - Can be build with eg. an LM317 and a bit of "glue".
Also is a single sell enough to power THIS? I read the data sheet but found it confusing.
Read page 6.
Output Voltage (@ Iout = 0 to 800 mA and Vin = 4.75
to 10 V) 3.235 to 3.365 V
Vd Dropout Voltage
IO = 100 mA 1.0V (typ) 1.1V (max)
IO = 500 mA 1.05V (typ) 1.15V
IO = 800 mA 1.10V (typ) 1.2V (max)
That means that if you need 3.3V out at 500mA, you need 4.45V minimum in (worst case)
And your Lithium battery will go down to 2.5V..3V over its discharge life - remember to have an automatic switch off before it goes too low!
You need something like a buck/boost, SEPIC or Cuck regulator to pull it off. This will also be the most lean and green way to go.