It seems like you could set up a webhook to trigger a Travis CI rebuild upon a new star event [0] . I'm pretty iffy on this, any ideas?. You probably need some third-party service to trigger the rebuild [1]. Obviously running your own server would make this much easier.
https://github.com/no13bus/ohmyrepo is one such example. From there it would be straightforward to generate and deploy the new README.md from Travis CI [2].
[0]: https://developer.github.com/v3/activity/events/types/#watch...
[1]: https://docs.travis-ci.com/user/triggering-builds
[2]: http://krzysztofzuraw.com/blog/2016/automatic-blog-deploymen...
