Updating address bar with new URL without hash or reloading the page
You can now do this in most "modern" browsers!
Here is the original article I read (posted July 10, 2010): HTML5: Changing the browser-URL without refreshing page.
For a more in-depth look into pushState/replaceState/popstate (aka the HTML5 History API) see the MDN docs.
TL;DR, you can do this:
window.history.pushState("object or string", "Title", "/new-url");
See my answer to Modify the URL without reloading the page for a basic how-to.
Changing only what's after hash - old browsers
document.location.hash = 'lookAtMeNow';
Changing full URL. Chrome, Firefox, IE10+
history.pushState('data to be passed', 'Title of the page', '/test');
The above will add a new entry to the history so you can press Back button to go to the previous state. To change the URL in place without adding a new entry to history use
history.replaceState('data to be passed', 'Title of the page', '/test');
Try running these in the console now!