A good boost converter will be ~90% efficient (it can be more efficient, but not miniaturized like the pictures show). So, to begin with, you'll discard ~10% of your energy.
Now look at AA alkaline discharge curves: http://www.powerstream.com/AA-tests.htm
Indeed if you gadget reports the batteries as dead at 1.25V, you can still get ~40% more usage. But most gadgets don't work that way. 1.25V is the normal voltage of a fully charged NiMH rechargeable AA, so if you gadget works with those, the real cutoff point is likely closer to 1.15V or so.
The booster will boost the voltage (at the expense of current, obviously, plus losses) so that even though the battery will be below 1.15V, output voltage will be at 1.5V. Looking at the graphs, this buys you less than 1/5th, and if you consider the losses in the boost converter, probably half that.
The "800%" claim seems totally bogus. Perhaps with a gadget that stops working when the voltage falls below 1.45V per cell.
I'm not saying it's not a useful gadget (and I'm very impressed by the miniaturization: where is the inductor? This thing must run at several MHz at least!) — just that the claims should be taken with a large grain of salt.