###The Dash Button
The [Amazon] Dash Button operates at 3.3 V boosted from the battery’s nominal 1.5 V, drawing 200–300 mA from the battery when on and 2.3 μA when in sleep. This means the ~1200 mAh battery should be able to power the device for at least four hours while on and decades while in sleep. Since the button is only on for a few seconds when activated, it can probably be used close to 1000 times before the battery dies. Thus, the button should become obsolete long before the battery is depleted.
[It was] pointed out that U1 is probably a TI TPS61201 boost converter; the footprint, package markings, and pinout seem to fit.