How dangerous is it to refreeze meat that has been thawed?
In theory you could thaw and refreeze as many times as you like, though the changes in temperature would definitely alter the quality of the meat's taste and texture.

What matters most is how long the meat has been in the so-called "danger zone" speaking from a temperature perspective. The "danger zone" is defined as being between 41 to 135 °F (5 to 57 °C).

Here is an excerpt from The Professional Chef, by the Culinary Institute of America:


  Foods left in the danger zone for a period longer than four hours are considered adulterated. Additionally, one should be fully aware that the four-hour period is cumulative, meaning that the meter starts running again every time the food enters the danger zone. Therefore, once the four-hour period has been exceeded, heating, cooling, or any other cooking method cannot recover foods.