The first pedals
Looking back we will find out that the first guitar effects did not have a power supply input. Most of the pedals have been running on a 9V Block. Just like the first Wahwahs, Fuzz Faces and Booster.
Under the label Sovtek EHX produced even in the 80s and 90s Big Muff pedals that could only have been run on battery.
Of course there were some exceptions and therefore some of the pedals have been produced with two 1.5V batteries or an own power supply.
While the plugs of power supplies for synthesizer equipment mostly have positive tip polarity guitar equipment commonly use power supply plugs with negative tip polarity. You may have heard rumors according to which this happened because Boss wanted improve the sales of their own power supplies. But surely, rumors sometimes are only rumors….
18V or more?
In the beginning inventers maybe did not think about the possibility of running pedals on more than 9 Volt. But on the actual pedalboards you can find many studio quality pedals and therefore you will perceive even subtle differences.
The legendary Klon Centaur theoretically runs also on 9V but internally this pedal doubles the current voltage to 18V. And in the instructions for use of some other manufacturers (f. e. Fulltone, JHS, Xotic) also tell about the use on 18V.
But what does a higher current voltage do?
The sound will become different: it will become more open and less compressed. The midrange will be reduced while treble is emphasized. The dynamic increases. Not everybody like this kind of sound and therefore go on running their pedals on 9V. Nevertheless an overdrive pedal that runs on 18V is able to create a pretty nice sound.
Where is the limit?
18V are still not the maximum of power for guitar pedals. Some recording studios use devices running on 24V.
Hudson Electronics Broadcast is an overdrive pedal that is based on a mixing console and that accepts 24V. This has a decisive influence on the sound by freshening up the guitar signal and making it punchier. And also the sound of the legendary Deluxe Memory Man became really good on 24V.
Maybe it is worthwhile to run even modulation pedals, multi effects or digital delays on more than 9V. At a time where big pedalboards use high quality multiple power supplies technically there would not be any problems.
But it is important to read the instructions of use or ask the producer before using another power supply for a particular pedal. As I told you before a Klon Centaur doubles the current voltage internally what means that you would destroy this legend by powering it with 18V. Considering the current prices this might become a really annoying story.
Accordingly applies: well thought out experiments can be worthwhile but especially with OP amp overdrives you should be quiet careful.