Feedbin is a simple paid web-based RSS reader with a well-supported API.
Ranked in these QuestionsQuestion Ranking
Pro Clean UI
Uncluttered and intuitive to use.
Pro Classy interface
The web interface is nice, and the apps give a great native experience regardless of what platform you're on.
Pro Full API
Feedbin has a fully-featured, free, REST-style API.
Pro Free and open source
Licensed under MIT. Code available on GitHub.
This won't disappear in a year or two if ad revenue falls off.
Pro Supports JSON Feed
JSON Feed is a new syndication format that uses JSON instead of XML to store its data. JSON is generally considered easier to read and write than XML, making for less error-prone data structuring and broken feeds.
Con No Team Integration
No ability to share information with others within the native app.
Con There is no free plan
One of the few news readers that does not offer a free plan. They have a 14 day trial, after that a subscription is $3/month.