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According to the official blog, YouTube is serving "well over a billion views a day". That's at least 11,574 views per second, 694,444 views per minute, and 41,666,667 views per hour. Imagine the web server farm that can withstand that much traffic, and then remember that YouTube is just one of Google's many properties.
As we wrote earlier this week, YouTube still may not be making enough money to justify the 1.65 billion dollars Google has paid for it, but it definitely makes it up when it comes to visitors.