Is it possible to find out via Google Analytics the highest number of simultaneous visitors at any given time?
As Samwise has said, it's entirely impossible to see the number of simultaneous visitors historically using Google Analytics.
Your only two options (if you insist on using GA) are to use the live view tab which only shows data from the past 30 mins (not ideal) or simply report on the number of visitors per hour in standard reporting (again not exactly what you're looking for)
I have never used the service that Samwise linked to but it's worth a shot!
You can estimate that using this formula:
Hourly visitors can be easily found in Audience Overview and Average Time Spent on Site through "select metric" on the same view.
I've had a really quick look through analytics - and I can't see any way to filter through old "real-time" data that's available.
For what you want to achieve, think about using something like http://whos.amung.us/
Perhaps a Google Analytics guru could shed some light on what is, or isn't possible - but from where I stand, you'd be better off using the link supplied.
Hope that has helped,