Lighting is not as simple as I wish it was. The electrical part isn't too bad though. In order for a light to be "bright", you need an LED that is putting out a lot of lumens, and an optic that places the lumens where you want (a narrow beam in this case). If you start with the LED, many lights are using Cree's XM-L or XP-L LEDs. These LEDs max out at 10 watts and about 1000 lumens. Most optics waste a bunch of those lumens, so don't get too caught up in the absolute lumen numbers. 10 watts is the key number here. To get 10 watts out of 3 AAA cells (4.5 volts), you need to draw 10/4.5 amps, or about 2.2 amps. AAA alkaline cells are not rated for this. I can find numbers for Duracells for 1 watt each (3 watts for our 3-cell setup), where they are rated for about 20 minutes. Alkaline cells have high internal resistance, so while they may last a long time at low current, they heat up and get less effective at high current. Rechargeable Lithium cells like 18650s are made for this though. An average cell is rated for 7 amps discharge and other cells will do over 20 amps. To get 10 watts out a single 18650 (3.6 volts), you need 10/3.6 amps, or about 2.8 amps. A standard 3200mAh 18650 cell will put out 2.8 amps for about 65 minutes.
Mid-term summary - 3xAAA will struggle to do 3 watts for 20 minutes, 1x18650 will easily do 10 watts for 65 minutes.
The AA battery is a better option. The Duracell AA coppertop is rated at 1 watt for about 80 minutes. It's still not rated for anything over 1 amp draw though.
The C battery gets closer. The Duracell C coppertop is rated for 2 amps for about 50 minutes, so I extrapolate 2.2 amps for about 45 minutes.
So 3xC cells will do 10 watts for about 45 minutes, 20 minutes less than a single 18650.
Now for the part that annoys me. You can't just buy 18650 cells from Amazon and know that you are getting good stuff. Buy from liionwholesale.com, mtnelectronics.com, or illumn.com. There are other good retailers if you want to search through budgetlightforum.com, but these 3 are my go-to sites. I get protected cells so that I don't have to worry about killing batteries. Others get unprotected cells so that the light works as long as possible without going dark.
So if you really want to stick with alkalines, use 3xC batteries for a bright light, or 3xAA batteries for a dimmer light. If you want a bright light in a small package, then 1x18650s are the way to go.