I've made a Firefox extension to display updated chapters count in badge, you can get it from here. Reading list should obviously be configured to display updates. Feel free to submit requests or bug reports, but I don't want to deviate from original purpose too much.

Nice idea! (Possible) bug I encountered:

• The icon only displays updated chapters on the first page of the reading list, so it would work perfectly fine if I set my list to "Show All" releases, but if I choose, say "Show 25" and if my list exceeds that number (which it does by far... ) then only the first 25 titles will be checked.

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Thanks for report, I'll try to fix this. No proper API really sucks balls, everything is more complicated with page scraping.