I know Andrew's swamped, but just so this is noted to the brains-that-be.
The topic pretty much says it all. The asterisk never goes away from the tab in Google Chrome at least. Once there has been a new chat, the new-chat asterisk will not go away no matter how thoroughly you check the place.

I've noticed this. When I am in a different window, I'm in chat in another window, and I go back to check the chat, and no one has said anything and so I can never tell if anyone has said anything when the sound is off because the * is always there.