A duplicate of this question came up on meta.SO where I suggested a solution, copied here:

My Temporary Hack

// sample script to generate output
var values = [
  ["a", "b", "c"],
  function foo() {
    return "bar";
  },
  document.createElement("a"), 
  { foo: "bar" },
  false,
  "foo",
  new Date()
];

// log each value with a delay
values.forEach(function(v, i) {
  setTimeout(function() {
    // call our custom log method... logging to both the DOM and the console
    console.log("i: %d, type: %s, value: %o", i, typeof v, v);
  }, 500 * i);
});

// throw an error
setTimeout(function(){
    throw new Error("test!");
}, 3500);

<!-- Provides the `snippet` object, see http://meta.stackexchange.com/a/242144 -->
<!-- To show result in the DOM instead of console, only to be used in this code snippet. Remove before using elsewhere. -->
<script src="http://gh-canon.github.io/stack-snippet-console/console.min.js"></script>

Notes:

• All calls to console.log() (info(), warn(), error(), and clear()) are passed along to the actual console
• Loosely imitates Chrome's output (colors and styles excluded)
• HTMLElements will be printed as their outerHTML
• Objects are printed using JSON.stringify() with an Object.prototype.toString() failover
• Works by injecting elements into the page... so, it obviously pollutes the DOM.
• The number of entries is limited to 50 by default. You can modify this limit by calling console.config({maxEntries:25}). Older entries will be removed to make room for newer entries.
• Logs unhandled errors.

<sub>- Minified version available here: https://gh-canon.github.io/stack-snippet-console/console.min.js
- Unminified version available here: https://github.com/gh-canon/stack-snippet-console/blob/master/console.js</sub>

A Real Solution

While my temporary hack above certainly works... what I'd really like to see is something which:

1. Is truly integrated, i.e.: doesn't link to my external script
2. Keeps the console's DOM separate from the snippet's DOM
3. Allows the author to opt in or out of the console wrapper

If we nest the snippet another frame down we can keep the console DOM separate from the snippet.

Page
 ┕ Console Frame
   ┕ Snippet Frame

Here's a demo. Click the button to log some mouse events.

The console page must have access to the snippet's window before we can overwrite the snippet's console. So, you'll need to inject the following into the snippet (as early as possible) in order to intercept subsequent calls to console.log():

<script>window.parent.attachConsole();</script>

Opting in/out?

<!-- begin snippet: js hide: false console: true -->

That could also control whether or not we need to nest the snippet in a containing frame for the console.