How can I highlight the line of text that is closest to the mouse?

Not sure if jQuery will help you out much here, but you could take a look at the element.getClientRects method, documented on MSDN and MDC. More specifically, this example at MSDN is sort of similar to what you want to achieve, highlighting lines using a cleverly z-indexed div element that goes behind the text at the co-ordinates returned by getClientRects().

You should be able to achieve the same thing by looping through the TextRectangle objects returned in the document's onmousemove and checking to see if the y value of the mouse cursor is > the top and < the bottom of each rectangle and moving the cleverly z-indexed div to the same position/height.

All the current major browsers support getClientRects().

http://jsbin.com/avuku/15

UPDATED - working in Chrome, IE6/7/8, Firefox, Opera, Safari. The initial problems I had in the other browsers were related to the DIV needing to be display: inline.
UPDATED AGAIN - I had to refer to this answer for some newer questions, so I took the time to update it to recalc the lines on window resize. It looks like others have been playing around too, it's now on revision 15.

I don't see how you could feasibly do this without explicitly-wrapped text (i.e., newlines or <br> elements).

To the best of my knowledge, there's no way for the DOM to discover where a specific piece of text has wrapped, character-wise nor pixel-wise - including what I know of the Range API - not to mention the dynamic nature text can assume, such as with the text-zooming feature of browsers.

But if you could somehow manage to generate/inject explicit line-endings, then I think I have a solution for you.

EDIT

Thanks to the awesome information in Pekka's answer, I've cobbled together a functional prototype, but it has a significant caveat - works with plain-text content only. Any HTML present the body of the element will be stripped.

jQuery.fn.wrapLines = function( openTag, closeTag )
  {
    var dummy = this.clone().css({
            top: -9999,
            left: -9999,
            position: 'absolute',
            width: this.width()
        }).appendTo(this.parent())
      , text = dummy.text().match(/\S+\s+/g);

    var words = text.length
      , lastTopOffset = 0
      , lines = []
      , lineText = ''
    ;

    for ( var i = 0; i < words; ++i )
    {
      dummy.html(
          text.slice(0,i).join('') +
          text[i].replace(/(\S)/, '$1<span/>') +
          text.slice(i+1).join('')
      );

      var topOffset = jQuery( 'span', dummy ).offset().top;

      if ( topOffset !== lastTopOffset && i != 0 )
      {
        lines.push( lineText );
        lineText = text[i];
      } else {
        lineText += text[i];
      }

      lastTopOffset = topOffset;
    }
    lines.push( lineText );

    this.html( openTag + lines.join( closeTag + openTag ) + closeTag );
  };

  $(function()
  {
    $('p').wrapLines( '<span class="line">', '</span>' );
  });

span.line {
  display: inline;
}
span.line:hover {
  background-color: lightblue;
}

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<p style="max-width:400px">
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</p>