The Opalec NewBeam is a fully regulated device. Driving it with higher voltages should not make very much difference. But what about driving it with lithiums, rechargables, or those 3.6V memory lithiums?
Let's make some quick ballpark calculations. 3 LEDs standard Nichia Whites. Assume each is driven at 4V and 40mA. Consumes 160mW each or 480mW total.
Assume converter/regulator circuit is 80% efficient. Power drain from batteries is 600mW.
Battery Voltage Current Required
--------------- ----------------
2.4V 250mA
3.0V 200mA
3.4V 176.47mA
3.6V 166.66mA
4.8V 125mA
7.2V 83.3mA
As can be seen, the current draw is not too severe, though it isn't light either. Using the Energizer L91 with their nominal current of 1.7V each, flat discharge and low internal resistance should help a little, but there probably won't be any dramatic improvements.
Rechargables with their low internal resistance should have little trouble dealing with a mere 250mA, but you can probably put 3 of them together and get an improved runtime. Heck, see if you can adapt the NewBeam to fit those Energizer DoubleBarrels that take 6 batteries (to get 3.6V). You could even take 3 alkalines or 4 NiMH for 4.5-4.8V.
However, even though Opalec suggests that you can input 7.2V with little detriment to the circuitry, there is little reason to try. The 3.6V lithium batteries in AA configuration are meant for backup power for volatile memory. They are only rated to deliver around 20mA constant. So they can't deliver the power required to drive the NewBeam properly.
You could use a battery pack that allows 6 NiMH batteries to be used, and you would probably improve the runtime somewhat, but you would be at risk of destroying the circuitry, since input voltage is an important parameter to the proper running of the circuits. A better option if improved runtime is required is to use parallel batteries configuration at the recommended voltage. This has 2-fold advantage in that a larger store of power is available and that the parallel configuration reduces the current draw on each cell, reducing the effects of internal resistance.
But since the NewBeam is fully regulated, increased voltage will not increase brightness.