Oh man, I wish the world's national leaders would believe as much in the free movement of people as they would of the free movement of stuff and capital.
The EU passport is certainly handy because it opens up a large part of the developed world to live it.
The alternative approach is to be rich. It's more than ERE standards, but about $500k will give you access to countries through start-up business visas. Some countries charge less, for the US it's around half a mil. I even remember seeing some organization in the US that was running some LLC where you bought a 500k share---it has to do with "creating jobs for Americans".
The second alternative approach is to be highly educated. That's the H1 or J1 equivalent of the US. Of course that ties you to a company instead. This transnational corporation domain is probably where the world is moving anyway.
Then there's the WTSHTF reason which is escaping a country gone crazy. This would likely be done along with a bunch of other refuges and having an alternate passport to separate yourself from the crowd would help. (I really think North America is one of the least likely regions in the world to call for this problem.) In this regard, I'm not sure a passport from an alternative trouble-zone is going to be as strong as a trouble-free country.
The downside to dual citizenship, I'm told, is that if you're ever in diplomatic trouble, each country might say that you're the "other country's" problem. Also more paperwork wrt taxation for some countries.