I have a custom piece of Javascript which I would like to run on every web page from specific domains, or perhaps simply on every web page.
(If you are wondering: i
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Greasemonkey should do what you want. If it's not working either it's not being applied to the correct domains, or the code contains some sort of bug! (I personally use grease monkey on stack overflow to make some changes to the answer area).
Try placing some alerts() within your code to ensure that your grease monkey script is executing as intended.