A Beast Cannot Feign (Gaia's Wasp Part 3) mentions that opening the gate of a power mosfet required the departure of three hundred billion electrons.

Actually the book said transistor, but it meant power mosfet. This is okay.

A mosfet gate is effectively a tiny capacitor, which must be charged to open the mosfet channel. That is the whole point of a power mosfet, to control current by opening or closing the channel.

Charging the gate (and hence opening the channel) consumes power, so the less the better and mosfet designers work hard to minimize the total gate charge.The required gate charge varies with mosfet size, but in 2014 a typical power mosfet requires a gate charge on the order of 50nC (nanoCoulombs.)

One Coulomb is 6.21018 electrons, so a gate charge of 50nC = 6.21018  5010‑9 = 31011 electrons.