Want to know more about your followers on Twitter and how influential they are? Fruji is a webapp that offers extensive analytics on your followers.
Basic analytics are free; if you spring for the premium or pro packages you'll get additional information such as identifying new followers, spam followers, and finding out what users have unfollowed you. The app also tracks your mentions and follower demographics.
You'll need to give Fruji read permission of your Twitter account and your email address, but the webapp doesn't try to access any additional privileges. If tracking social media penetration is important to your business it might be another weapon in your arsenal.