Google Map V3, how to get list of best driving route for multiple destinations?

The link you refer to explains how to do this: admittedly you also get back some directions information you don't care about, but you can simply ignore it.

To be clear, you just make a directions request with some waypoints (say B,C and D) in addition to your origin and destination (A and E).

If using the web service, ensure you set optimize:true before listing your waypoints, and check the waypoint_order array (near the bottom of the result for this url) to get the info you want:

http://maps.googleapis.com/maps/api/directions/json?origin=Adelaide,SA&destination=Adelaide,SA&waypoints=optimize:true|Barossa+Valley,SA|Clare,SA|Connawarra,SA|McLaren+Vale,SA&sensor=false

If use the JS API, set optimizeWaypoints:true in your request.

As always, be sure you're abiding by the terms of service (e.g. results from your web service request must be displayed on a Google Map)

You might want to look at the Distance Matrix service: https://developers.google.com/maps/documentation/javascript/distancematrix

It gives you distances between a set of origins and destinations, and might help you with narrowing down options.