Thanks for asking 
I retrieve some of the following information through custom Data Explorer queries. I consider these metrics relevant for our use case.
Users
- number of registered users in group X
- share of contributing users (created a topic, replied to a topic), reading users, and ghost users (just clicking around without further engagement)
- language setting (from the language switch)
- distribution of email domains (public providers, universities, others)
Topics, Posts
- number of topics / posts in self-defined groups of categories
- post-per-topic ratio in these groups
Traffic
- page views from web crawlers
- page views from users = anonymous users + logged-in users
- clicks from Top 5 referrers
Community Health
- DAU/MAU ratio
- number of daily engaged users (DEU)
- share of TL-1 users in group x