"I think AA vs. AAA is an interesting question."
Let's start with the assumption that the maximum current from one AA is equivalent to the maximum current from 3xAAA (but that has to be with them in parallel). If those are the same, then the only advantage that 3xAAA offers is that the 3xAAA has the batteries in series, so you have a higher voltage available to drive the LED directly and not have to fool around with the PWM chip to boost the AA voltage. So, the 3xAAA solution (with no chip) might have the advantage of less cost and a teeny tiny bit of weight savings. OTOH, if you have to go through a chip anyway in order to get fancy dancy multilevel intensity, blinking modes, and better regulation, then it washes.
For purchasing batteries over time, it is probably cheaper to get one AA rather than to get 3xAAA. Where 3xAAA gets tricky is that those batteries are not as commonly stocked in remote stores. Where that gets really tricky is if you try to get lithium primaries, because lithium AAA batteries are very hard to find in some areas.
The good news is that lithium batteries run for a very long time, so replacement is not very often. The bad news is that lithiums have a sharp discharge knee, so when they start to fail, they fail completely in a hurry. But, that applies to all lithium primaries, AA or AAA alike.
If I really want something to run a long time, I use my Pac-Lite. That is a 9-volt lithium battery with a miniscule head on top. The head has a tiny switch for High/Low/Off and two LED emitters. No shell around the battery. No reflectors. No head straps. No frills. That sucker runs forever.
–B.G.–