An LED's brightness is determined by the CURRENT flowing through it. You can run it near its maximum current rating and minimise fluctuations by using a simple (one transistor) current regulator.
You can make it look even brighter by supplying it with a train of high current pulses, such that the average current does not exceed its maximum rating. The beauty of the Arduino is that you don't need to build a fancy pulse generator. You can do it in the firmware! Just make sure that the driver transistor can cope with the current pulse. For example, a ZTX650 with a 10 Ohm series resistance to limit the maximum current would be a starting point. Just make sure that it's impossible for the LED to remain on. It must always be pulsed. (Refer to the simple LED flasher example.)