JSFiddle

If you want to edit some complex javascript I suggest you use JsFiddle. Alternatively, for smaller pieces of javascript you can just run it through your browser URL bar, here's an example:

javascript:alert("hello world");

And, as it was already suggested both Firebug and Chrome developer tools have Javascript console, in which you can type in your javascript to execute. So do Internet Explorer 8+, Opera, Safari and potentially other modern browsers.

You can use a light weight webserver to serve the file.
For example,
1. install Node
2. install the "http-server" (or similar) package
3. Run the http-server package ( "http-server -c-1") from the folder where the script file is located
4. Load the script from chrome console (run the following script on chrome console

var ele = document.createElement("script");
var scriptPath = "http://localhost:8080/{scriptfilename}.js" //verify the script path
ele.setAttribute("src",scriptPath);
document.head.appendChild(ele)

1. The script is now loaded the browser. You can test it from console.