Building on Eugen's answer, I wanted something self-contained (no id's, as inline as possible) that would not require a hosted pixel.gif image. I came up with a few possibilities:
One option would be to use a base64 encoded src URL instead (as small as possible). Note that data-uris are supported in IE8+:
<img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" onload="this.onload=null; this.src=getImagePath();">
See it in action on jsfiddle.
A second option would be to use document.write to write the image tag directly with an inline script. Just put the below <script> instead of the <img>. Note that many consider document.write to be bad practice:
<script>
  function getImagePath() {
    return "http://i.imgur.com/4ILisqH.jpg";
  }
  document.write('<img src="'+getImagePath()+'">');
</script>
See it in action on codepen.io.
A third (perhaps even more hackish) option would be to forcibly break the image by supplying a null src, and (ab)use the onerror callback.
This works in IE11 and Chrome, but not Firefox:
<img src onerror="this.onerror=null; this.src=getImagePath();">
See it in action on jsfiddle.
This fourth option relies on a simple function that sets an <img> tag followed immediately by an inline <script> that sets the image's src via JavaScript:
<!-- In the <body> -->
<img><script>set_most_recent_img_src(getImagePath())</script>
And in your <head>:
<!-- In the <head> -->
<script>
  function getImagePath() { return "https://www.gravatar.com/avatar/71ec7895cada78741057c644d858b0e3"; }
  function set_most_recent_img_src(val) {
    var a = document.body.getElementsByTagName('IMG');
    var img = a?a[a.length-1]:0;
    if (img) img.src = val;
  }
</script>
See it in action on codepen.io.
Summary: I'm just ideating. Each option has different implications and requirements -- they're all workarounds, and should be tested thoroughly for your specific use case. Personally I think the first option (base64 URI) is the most solid (if you ignore old IE browsers, which I happily can).