I dug into this a bit and found some CSS that will remove it.
I do not suggest this because it will be cutting yourself and all of your users off of our live support chat which I have found very very successful. A user has an issue, shoots us a message, and we can fix it in a matter of minutes, much faster than our previous support methods.
Here's the code if you do decide to hide it!
display: none !important;
Edit: After thinking a bit more on it, I'm not exactly sure what this will do to the automatic messages that will pop up. Not exactly sure how to test that either. The code I provided here definitely gets rid of the message bubble at the bottom right.