1 single 4.2v (Fully Charged) lithium battery will supply max brightness (in the flashlight world they say “direct drive”, most would say a short i guess lol) …
BUT, that 4.2v soon becomes 4.1…3.8 in less than a few seconds, your current as a consequent drops… the best way is to keep that current up is to build a driver, sadly most people in the electronics
think of super bright LED’s as anything under 30ma lol… i’m dealing with 5watt and 10watt RGB and 15watt white LED’s which all run fine at 4.2volts without a resistor (except for the 12/24vdc LED’s)
but you need a resistor if you wish the arduino to power it (unless you want to damage the pin/processor). if you wish to power you need a suitable power supply, a power transistor or a mosfet and a big heatsink, also you may prefer to use a PWM pin and use analogWrite on it with say a power transistor and gradually dim/light it (as i do with a 12watt 4.2v LED/TIP31,for that, i run it at around 0.9amps or Full 1.3amps, but i use a 300 ohm resistor on base to protect the pin on the arduino, i could go a lot higher with another method)
your power supply you plug into the wall does not deliver like a battery, chances are give it 4.7volts and you will see smoke, batters can only supply 4.2v, this is why manufacture specifications are important.