How can I suppress chrome-extension:// output from Web Inspector's console log?

Firefox has a way to change the console logging level for specific extensions from about:config.

Chrome doesn’t have this capability. Your options:

  1. Filter out all messages from a particular source file by right-clicking the white (blank) part of console message line and choosing Filter > Hide Messages from foo.js. You can undo this filtering from the same menu later, if you wish. Chrome will remember to do this filtering for you until you disable it (even if you navigate somewhere else or restart Chrome).
  2. Override one or more console methods. This approach only affects the current scripting context. Chrome extensions run in their own isolated worlds with their own window objects, global scopes, and console objects. If you can’t edit the source code of the extension, you’ll need to paste and run your console-altering scripts in each extension’s execution context, and your modifications will only be in effect until the page is unloaded. You can change the console’s effective execution context using the dropdown at the bottom of the console that says <page context>. Note that any frames or iframes in the page also have their own execution contexts, accessible using the frame dropdown (initially set to <top frame>).

Just temporarily disable the extensions.

This will add a button to your browser and it will only require two clicks to toggle your extensions.