I’m afraid this one is for those with a bit of spare cash, but a number of countries allow you to buy citizenship. One of the cheapest is Dominica (not to be confused with the Dominican Republic), a tiny island nation in the Caribbean.
Dominica is famed for its pristine, natural beauty: its jungles are virtually untouched, its beaches warm and plentiful, and its citizenship is up for sale.
The cost is $75,000 US for an individual or $100,000 for a family. Yes, that link goes to a commercial firm which offers “assistance” for this service, but the process is quite legal and fairly well known. Simply searching for Dominica Naturalization and Citizenship Act will generate plenty of hits for you to research.
The capital of Dominica, Roseau, has fewer than 15,000 people, compared to less than 75,000 for the entire country. The Dominica per capita GDP is a little over $10,000, though I’ve had trouble finding reliable figures for the median income (it appears to be around $5,000 per year). If you don’t mind poverty, being in the middle of nowhere and gorgeous natural scenery, Dominica offers you that second passport.
For a more human touch, here's an interesting blog on an American who moved to Dominica and subsequently left. Note that the author is pretty upset with the Dominica government and has no plans to return there.