Can I change favicon without modifying the master page?

The another option of changing the favicon icon without modifying master page is using jQuery.

Upload your favicon icon in SharePoint and use below jQuery code to replace this on run time:

$("link[rel='shortcut icon']").attr("href","[Favicon Icon URL]");

Hope this will help you meet your requirement.

In 2013, with server access you can just replace the 'favicon.ico' file in the _layouts/15/images/favicon.ico But I don't think there is a way without modifying the master page in SharePoint Online.

It is highly unlikely, that you will break something by replacing the path to the favicon in your master page. In addition, it don't believe there is a way without laying hand on your master page file.

So it is back to basics I'm afraid

1. Upload your custom favicon.ico file
2. Make a copy of your master page file.
3. Edit that copy and replace the path to the favicon.ico
4. Save & publish