I used to have site stats plugin wp-shortstat activated which enable to see how many visitors I have, where are they from and what have they clicked. I later found out that the stats plugin generated huge amount of records in the database which is not necessary. I turned it off.
I have also subscribed to free service of statcounter where I could put script in my sidebar to show how many unique visitors have been here. Recently that has stopped loading of the page right in the middle of it due to slow response of the statcounter site. Well, why do I care?
I have now moved it to the footer or even have it deleted later on.
This is not a commercial site and I don't really care statistics.