Depending on the nature of the chats, around a half dozen is my limit. If you've got 1-2 that are only responding to you every 10 minutes or so, or 1-2 have easy general questions, that number could probably push higher. But around 5 conversations that require legitimate investigation and troubleshooting is where I've found you can start to lose engagement.
I find I can handle multiple chats at once easily using programs like TextExpander(Mac) or Breevy(Win).
That way, if someone needs a specific walkthrough, I can type something like oguide and it automatically pastes this link: https://www.olark.com/help/guide
Then if you model your support system around making the walkthroughs and guides for the most common user issues, it makes juggling chats much easier.