There’s plenty of anecdotal evidence floating around the internet about musicians, specifically guitarists, only using batteries that were almost dead in their pedals to get the best sound. These anecdotes might not be 100% true, but there does seem to be some merit in using batteries that aren’t fully charged in certain pedals.
Using this battery technique will work best on transistor based analog circuits, and in particular fuzz and distortion pedals. The Dunlop Fuzz Face is one pedal where ‘sagging’ the voltage has a well-documented effect on the sound.
Digital pedals will not respond as well to using half dead batteries, as their modern circuits are designed to not be affected by different voltage levels. It’s recommended to stick to analog pedals when trying this method.
Batteries that are drained produce less voltage, and thus power the pedal at a voltage that is under the normal level required, which gives the pedal a different sound. In fuzz pedals, one theory is that the lower power sucks some of the gain out and adds a little more compression, giving the pedal a lo-fi sound.
It’s worth noting that as well as using half dead batteries that ‘sag’ the power, there are some power supplies out there which give this option, and there are even a few pedals on the market which do the same.
If you can’t quite nail the fuzz or distortion sound you’re looking for, it might be worth unplugging from your power supply and looking for some batteries being used around the house to put in your pedal (preferably not from the smoke detector).