I know that I can browse the chat when I'm not logged on. But what if I am logged on and want to spare the inconvenience of logging off/multiple browsers?
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Instead of jumping into the chatroom, you can view the transcript. In this example, the transcript is at http://chat.stackexchange.com/transcript/2496.
Reading the transcript does not count you as entering the room. While it will not update in real time, every time you refresh will update the transcript for any new messages. Not the most efficient means, but otherwise, there isn't any other means of silently observing a room in-engine.
Well, you can always monitor the transcript. It doesn't refresh automatically, but if you're not looking to be an active participant in the chat I'm not sure why this would be an issue.
Otherwise, just open a private browsing session in your browser of choice where you won't be logged into the Stack Exchange network, and access the room that way. It's a little extra work on your part, but this seems like an odd use case to me anyway.