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">
 one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty twenty-one twenty-two twenty-three
</p>

The best approach that comes to mind is splitting each line into a <span> or <div> element that has a :hover CSS class with the "highlight" setting set:

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

That would be the least expensive solution, as the browser is going to take care of all the highlighting itself. If you want fancy effects, you can still achieve that by adding mouseover and mouseout events to every line.

The tough part, of course, is splitting the content into lines at the browser's line break. You need to do that dynamically so the lines actually reflect the positions at which the browser breaks the text.

Maybe the accepted answer to this question is a step into the right direction:

Getting a specific line using jQuery

How it works:

It goes through the entire element (actually, a clone of the element) inserting a element within each word. The span's top-offset is cached - when this offset changes we can assume we're on a new line.