list every font a user's browser can display

Yes there is! I'm so glad you asked this question because I now want to use this too.

http://www.lalit.org/lab/javascript-css-font-detect

Code from http://www.lalit.org/wordpress/wp-content/uploads/2008/05/fontdetect.js?ver=0.3

/**
 * JavaScript code to detect available availability of a
 * particular font in a browser using JavaScript and CSS.
 *
 * Author : Lalit Patel
 * Website: http://www.lalit.org/lab/javascript-css-font-detect/
 * License: Apache Software License 2.0
 *          http://www.apache.org/licenses/LICENSE-2.0
 * Version: 0.15 (21 Sep 2009)
 *          Changed comparision font to default from sans-default-default,
 *          as in FF3.0 font of child element didn't fallback
 *          to parent element if the font is missing.
 * Version: 0.2 (04 Mar 2012)
 *          Comparing font against all the 3 generic font families ie,
 *          'monospace', 'sans-serif' and 'sans'. If it doesn't match all 3
 *          then that font is 100% not available in the system
 * Version: 0.3 (24 Mar 2012)
 *          Replaced sans with serif in the list of baseFonts
 */

/**
 * Usage: d = new Detector();
 *        d.detect('font name');
 */
var Detector = function() {
    // a font will be compared against all the three default fonts.
    // and if it doesn't match all 3 then that font is not available.
    var baseFonts = ['monospace', 'sans-serif', 'serif'];

    //we use m or w because these two characters take up the maximum width.
    // And we use a LLi so that the same matching fonts can get separated
    var testString = "mmmmmmmmmmlli";

    //we test using 72px font size, we may use any size. I guess larger the better.
    var testSize = '72px';

    var h = document.getElementsByTagName("body")[0];

    // create a SPAN in the document to get the width of the text we use to test
    var s = document.createElement("span");
    s.style.fontSize = testSize;
    s.innerHTML = testString;
    var defaultWidth = {};
    var defaultHeight = {};
    for (var index in baseFonts) {
        //get the default width for the three base fonts
        s.style.fontFamily = baseFonts[index];
        h.appendChild(s);
        defaultWidth[baseFonts[index]] = s.offsetWidth; //width for the default font
        defaultHeight[baseFonts[index]] = s.offsetHeight; //height for the defualt font
        h.removeChild(s);
    }

    function detect(font) {
        var detected = false;
        for (var index in baseFonts) {
            s.style.fontFamily = font + ',' + baseFonts[index]; // name of the font along with the base font for fallback.
            h.appendChild(s);
            var matched = (s.offsetWidth != defaultWidth[baseFonts[index]] || s.offsetHeight != defaultHeight[baseFonts[index]]);
            h.removeChild(s);
            detected = detected || matched;
        }
        return detected;
    }

    this.detect = detect;
};

Summary

How does it work?

This code works on the simple principle that each character appears differently in different fonts. So different fonts will take different width and height for the same string of characters of same font-size.

The JavaScript version is a bit flaky. It gets fonts by iterating through known fonts and testing.

The most accurate way (albeit having to use a propriety plugin) is to use Flash. Here you can get the list of fonts without having to test for them individually using dimensions.

You are going have to decide whether to have an exact list at the expense of not working on some devices ( iDevices, browsers without Flash plugin, etc), or a partial list with better support via JavaScript only.

There is a way to do this using document.fonts

The returned value is the FontFaceSet interface of the document. The FontFaceSet interface is useful for loading new fonts, checking the status of previously loaded fonts, etc.

• Returned values are verbose with weight, style, etc.

function listFonts() {
  let { fonts } = document;
  const it = fonts.entries();

  let arr = [];
  let done = false;

  while (!done) {
    const font = it.next();
    if (!font.done) {
      arr.push(font.value[0]);
    } else {
      done = font.done;
    }
  }

  return arr;
}

• Returns only the font family

function listFonts() {
  let { fonts } = document;
  const it = fonts.entries();

  let arr = [];
  let done = false;

  while (!done) {
    const font = it.next();
    if (!font.done) {
      arr.push(font.value[0].family);
    } else {
      done = font.done;
    }
  }

  // converted to set then arr to filter repetitive values
  return [...new Set(arr)];
}

I have tested it without linking any fonts in the HTML, then linked Roboto font, tested again and it got added to the result.