Is it possible to append to innerHTML without destroying descendants' event listeners?

Unfortunately, assignment to innerHTML causes the destruction of all child elements, even if you're trying to append. If you want to preserve child nodes (and their event handlers), you'll need to use DOM functions:

function start() {
    var myspan = document.getElementById("myspan");
    myspan.onclick = function() { alert ("hi"); };

    var mydiv = document.getElementById("mydiv");
    mydiv.appendChild(document.createTextNode("bar"));
}

Edit: Bob's solution, from the comments. Post your answer, Bob! Get credit for it. :-)

function start() {
    var myspan = document.getElementById("myspan");
    myspan.onclick = function() { alert ("hi"); };

    var mydiv = document.getElementById("mydiv");
    var newcontent = document.createElement('div');
    newcontent.innerHTML = "bar";

    while (newcontent.firstChild) {
        mydiv.appendChild(newcontent.firstChild);
    }
}

Using .insertAdjacentHTML() preserves event listeners, and is supported by all major browsers. It's a simple one-line replacement for .innerHTML.

var html_to_insert = "<p>New paragraph</p>";

// with .innerHTML, destroys event listeners
document.getElementById('mydiv').innerHTML += html_to_insert;

// with .insertAdjacentHTML, preserves event listeners
document.getElementById('mydiv').insertAdjacentHTML('beforeend', html_to_insert);

The 'beforeend' argument specifies where in the element to insert the HTML content. Options are 'beforebegin', 'afterbegin', 'beforeend', and 'afterend'. Their corresponding locations are:

<!-- beforebegin -->
<div id="mydiv">
  <!-- afterbegin -->
  <p>Existing content in #mydiv</p>
  <!-- beforeend -->
</div>
<!-- afterend -->

Now, it is 2012, and jQuery has append and prepend functions that do exactly this, add content without effecting current content. Very useful.