How to connect any external USB drive to XBOX360 and use all its capacity above 16gb

Most of console users know that Microsoft gracefully allowed their customers to use USB drives as Xbox360-enabled storage.
However, if your old good NES or Sega had their limitations justified by inferior hardware, modern consoles are so powerful that marketing guys have to impose artificial limitations to force consumers buy proprietary products. That is official Xbox360 hardrives, which cost 3 times more than the very same Hitachi/WD drives they are rebranded from. On non-proprietary USB drives, only 16gb of space can be used by the console, the rest is wasted.
What to do if you happen to get a much bigger drive and want to use its full capacity? Well, here is how.