There was another thread about this, which I've tried. But there is one problem: the textarea doesn't shrink if you delete the content. I can't find any way to shrink it to the correct size - the clientHeight value comes back as the full size of the textarea, not its contents.

The code from that page is below:

function FitToContent(id, maxHeight)
{
   var text = id && id.style ? id : document.getElementById(id);
   if ( !text )
      return;

   var adjustedHeight = text.clientHeight;
   if ( !maxHeight || maxHeight > adjustedHeight )
   {
      adjustedHeight = Math.max(text.scrollHeight, adjustedHeight);
      if ( maxHeight )
         adjustedHeight = Math.min(maxHeight, adjustedHeight);
      if ( adjustedHeight > text.clientHeight )
         text.style.height = adjustedHeight + "px";
   }
}

window.onload = function() {
    document.getElementById("ta").onkeyup = function() {
      FitToContent( this, 500 )
    };
}