Which users actions are worth logging, to gain better perspective on their behavior?

I am currently implementing a "User action log", that will keep interesting actions of users within our system.

Any advice on what to log? Of course, the answer is "everything you feasibly can", but I want to focus on the important actions, that will help us gain true insights.

Log every button click? Every hover on a button? Same for hrefs? What else?

Note:

  1. The goal is better UX - we want to study our users, understand how they behave, and build a better system for them.
  2. I'm aware of some 3rd party solutions that track everything users do, starting from Google Analytics, through apps like Clicktale. I'm not saying we won't use those - but still, a custom log with some select actions might provide extra value over these "log everything" type of systems.
  3. The focus is websites, built for consumption over standard computers (mobile/tablet clients are less relevant for us a.t.m).