How Low Can You Go

For powering the Pico W, page 15 of the datasheet specifies a lithium cell or 3x AA batteries when away from USB power. It also says you should use a diode to prevent USB voltage from back-powering the batteries. 

We won't be doing either of those things. 

Doing more with less

1. We have no space in our button for three AA batteries. The Positivity Pusher is based on commonly available recordable buttons which have a battery compartment for two AAA batteries. Will that be enough to power the Pico? The answer is yes! 
2. That protection diode causes a voltage drop. Not ideal for maximizing lifespan, so it's omitted it in the final iteration. This does introduce the possibility of a....

SAFETY ISSUE!

If you plug the Pico into USB while also powering it from the batteries, the batteries will get really hot. If you don't notice right away, something worse might happen. If you're going back and forth between batteries and USB power, I strongly recommend leaving the diode in place until the coding is finished and the button is closed. Additionally, it might be a good idea to cover the Pico's USB port with a label reminding future-you to remove any batteries before plugging in. 

Power Hours

The Pico W needs a minimum of 1.8v. 

The Positivity Pusher's button compartment wires the batteries in series, which will deliver 3v using non-rechargeable batteries and about 2.4v with rechargeable batteries. 

A safety diode will introduce a voltage drop. Exactly how much depends on the type of diode, but many common types will drop .7 volts. That's enough to drop the voltage of rechargeable batteries to 1.7v, not enough for the Pico to operate. 

Runtime Expectations

A button should last at least a month on one set of batteries.

The capacity of two AAA batteries will be somewhere in the 2000mah range depending on type. Power consumption looks something like this:

After playing a clip and reloading the next one, the rp2040 processor will drop into deep-sleep mode, where it spends the majority of it's time. In this mode, power consumption should be a scant 1.4mA. 

While playing audio or downloading clips, power consumption can be as high as 100mA (including the indicator LED).

Total runtime is influenced by battery type, frequency of presses, and even distance from the wifi router. Using a battery life calculator, I estimate that with non-rechargeable lithium batteries the button should last more than two months, even with 5 button pushes per day. 

This page will be updated with real-world runtimes once they have been measured.