As the title states, the 1st-generation Amazon Fire TV Stick running software version 126.96.36.199 has been rooted for the first time. XDA forum member tehlers, along with the help of forum member Spider1996, have successfully used the Dirty COW vulnerability to root that specific Fire TV Stick model running that specific software version. The reason this rooting procedure only works on that specific device and software is because unlike most rooting methods which inject the SU binary into the existing operating system, this rooting procedure overwrites the entire system partition, which contains the operating system, with one that is pre-rooted. So the files used must be tailored made for each device model and software version.
I do not know if other currently unrootable Fire TV devices and software versions are susceptible to this rooting tactic, if someone were to tailor make files for them, but for now, it’s only available for the Fire TV Stick 1 running software version 188.8.131.52. If you have that specific device running that specific software version, you can download the necessary files and follow the procedure laid out in this post. Do not try to use the same files and steps on any other device or other software versions because it will very likely result in bricking your device.
I do not have a Fire TV Stick 1 running software version 184.108.40.206, so I am unable to test this procedure myself or write a guide for it. I have updated my rooting guide to include this new method, but you’ll need to have faith in tehlers and Spider1996’s work.