Can you determine if Chrome is in incognito mode via a script?

One way is to visit a unique URL and then check to see whether a link to that URL is treated as visited by CSS.

You can see an example of this in "Detecting Incognito" (Dead link).

Research paper by same author to replace Detecting Incognito link above

In main.html add an iframe,

 <iframe id='testFrame' name='testFrame' onload='setUniqueSource(this)' src='' style="width:0; height:0; visibility:hidden;"></iframe>

, and some JavaScript code:

function checkResult() {
  var a = frames[0].document.getElementById('test');
  if (!a) return;

  var color;
  if (a.currentStyle) {
    color = a.currentStyle.color;
  } else {
    color = frames[0].getComputedStyle(a, '').color;
  }

  var visited = (color == 'rgb(51, 102, 160)' || color == '#3366a0');
  alert('mode is ' + (visited ? 'NOT Private' : 'Private'));
}

function setUniqueSource(frame) {
  frame.src = "test.html?" + Math.random();
  frame.onload = '';
}

Then in test.html that are loaded into the iFrame:

<style> 
   a:link { color: #336699; }
   a:visited { color: #3366A0; }
</style> 
<script> 
  setTimeout(function() {
    var a = document.createElement('a');
    a.href = location;
    a.id = 'test';
    document.body.appendChild(a);
    parent.checkResult();
  }, 100);
</script> 

NOTE: trying this from the filesystem can make Chrome cry about "Unsafe Javascript". It will, however, work serving from a webserver.

The functionality of this answer is Chrome version dependant. The most recent comment was this works in v90

Yes. The FileSystem API is disabled in incognito mode. Check out https://jsfiddle.net/w49x9f1a/ when you are and aren't in incognito mode.

Sample code:

    var fs = window.RequestFileSystem || window.webkitRequestFileSystem;
    if (!fs) {
      console.log("check failed?");
    } else {
      fs(window.TEMPORARY,
         100,
         console.log.bind(console, "not in incognito mode"),
         console.log.bind(console, "incognito mode"));
    }

In Chrome 74 to 84.0.4147.135 you can determine this by estimating the available file system storage space

See the jsfiddle

if ('storage' in navigator && 'estimate' in navigator.storage) {
    const {usage, quota} = await navigator.storage.estimate();
    console.log(`Using ${usage} out of ${quota} bytes.`);

    if(quota < 120000000){
        console.log('Incognito')
    } else {
        console.log('Not Incognito')
    }   
} else {
    console.log('Can not detect')
}

You can, in JavaScript, see JHurrah's answer. Except for not highlighting links, all incognito mode does is not save browse history and cookies. From google help page:

• Webpages that you open and files downloaded while you are incognito aren't recorded in your browsing and download histories.
• All new cookies are deleted after you close all incognito windows that you've opened.

As you can see the differences between normal browsing and incognito happen after you visit the webpage, hence there is nothing that browser communicates to the server when it's in this mode.

You can see what exactly your browser sends to the server using one of many HTTP request analysers, like this one here. Compare the headers between normal session and incognito and you will see no difference.