If you're an inveterate doodler, all your notebooks are covered with drawings. If you're an inveterate golf player, you probably get twitchy if you haven't been out on a course in a week.
In Middle English inveterate was associated with chronic disease. Now it simply refers to something that is a signature habit with a person. Unless you're an inveterate gambler, drinker or smoker––in which case you're addicted and we're back to talking about being sick.