add my rank to my user page.
I'd like to know where I rank among the users with my rep score, so that I can see if I'm moving up or down in the world of SO.
a bit too explicitly game-like
Encourage it or not, it'll happen anyway (imho). I don't think it'd be so bad to have this.
You can do this fairly easily with the users list and a bit of math (it could be made easier though).
Jeff has mentioned before that he did not want to encourage competition with rep. Posting rank will probably be seen as encouraging competition.