Seriously people can be a bit paranoid about this. Ok if you thaw chicken breast or mince overnight at room temperature you would be daft to refreeze it. However, I have half defrosted pork chops to get a couple off a 'block' and refrozen the rest and, like retro, still have a pulse.
I've also half defrosted pies and pizza to cut in half and refreeze.
The reason the advice is not to refreeze is that while defrosting at room temperature bacteria can form which, if refrozen ,will remain on the food and if its then not cooked properly you may get sick. Very low risk.