I've always assumed that the quietest anechoic room belonged to either JBL (I was told that they have 3 of them) or the Institute for Research and Coordination In Audio and Music (IRCAM) in France, but according to the Guinness World Records, it's actually at Orfield Laboratories in South Minneapolis. Supposedly the Orfield chamber absorbes 99.9% of all sound generated within, which results in a measurement of -9dB SPL. As a comparison, a typical quiet room at night where most people sleep is at 30dB SPL, while a typical conversation is at about 60dB SPL.
The Orfield chamber is so quiet that no one has been able to stay inside for more than 45 minutes due to the fact that you begin to hear your heart beating, you lungs working, and even the blood coursing through your veins. Some people even begin to hallucinate during the experience. In fact, you can't even stand after a half-hour since you no longer hear the audio cues that you're used to when you stand as the reflections bounce off the floor, ceiling and walls of the environment.
While it's easy to figure out what JBL does with their anechoic chamber, what goes on in an independent one like at Orfield? It seems that the chamber is used by companies like Harley Davidson and Maytag to test how loud their products are. NASA also uses it for astronaut training.
Here's a short video that describes the Orfield anechoic chamber.