Usually I'll play something, and once I beat it (or all games if it's a series), I will totally immerse myself in the universe - read every article I can get my hands on, browse fanart, write fanfic, etc. Obviously the honeymoon period doesn't last forever, though!
Some games clearly have more staying power with me than others, as evidenced by my participation on this site.

Back in the day, I was highly obsessed with the
Myst series, but then real life got in the way. I haven't even played the last two games yet. (However, I can still tell you miniscule details from the accompanying novels since middle school-me read those over and over and over and over again!)