First of all i know it is a theoretical question and it contains no code but actually this time i don't have code, i need your suggestion on this issue.
Question:
Is it possible to fetch the name of all categories from a wordpress.com blog or a self-hosted blog connected through jetpack?
I tried finding it here https://developer.wordpress.com/docs/api/ but no luck.
So please suggest.
I've been looking for the same thing. I don't think this feature is available yet (As of 24th May 2013) on jetpack but available on hosted blogs (see the link below).
http://codex.wordpress.org/Function_Reference/get_categories
In case you are using JSON API plugin, you can use http://yourdomain.com/?json=get_category_index to get the list of all categories.
PS: You can use this JSON API plugin in place of jetpack api as it is hosted on your own blog with greater control.