I'm in the "running around" camp, not that I have any particular "home" to stay at.
I'm not a member of a club and play the vast majority of my golf on local, public courses. Earlier this year, based on some discussions with friends, I made a list of all the courses I'd played in Arizona as well as the courses I have not (at least around the metro Phoenix area). I've played 30 times this year, but have only played 2 courses twice (and one of those had undergone renovations). In all, I played 11 courses that I had never played previously and several others that I hadn't played in many years.
The thing of it is, even before this, I'd already played the vast majority of courses here in the Valley that I have much interest in playing--and my new experiences bore that out, for the most part. But I still really enjoyed seeing all the places I had not seen before.
Maybe that's just due to the relative quality of where I might have been playing otherwise. Again, no "home" course and good as Talking Stick or WeKoPa might be, for several reasons, it's not feasible that the majority of my rounds would be there if I hadn't been seeking out the never-before-seen's instead. Maybe that would color my thinking. I suppose if I could play Desert Forest every day, playing some housing development track in Mesa would be considerably less appealing. (But I might still do it.) Then again, Jason's original question is on the other end--running around the world playing great courses, or playing great courses at home.
I love seeing new golf courses. I don't have the resources to travel as much as I would like or play really great places every year, but I think the urge is the same--I love to see something new.