The end game will be inflation.
Show me an example in history where a paper currency gained in purchasing power as the issuing country went broke. It doesn't happen. What will happen, sooner or later, is a massive run on the U.S. dollar. I can't tell you exactly when it will happen, but I can tell you the signs of approaching trouble…
We're looking at many of them right now: Trading partners have begun avoiding the dollar for bilateral trade (Russia and China). The prices of gold and silver are soaring as central banks try to escape the U.S. dollar. Debt loads that can't be financed (in Europe, and U.S. banking system, U.S. homeowners) are creating massive economic problems. Prices for strategic commodities (copper and oil), foodstuffs, and farmland are soaring.
In the past (1979-1980, 1994), runs on the U.S. dollar were contained by increasing interest rates and reducing the growth of the Fed's balance sheet. This, in turn, reduced the money supply while increasing demand for dollars (thanks to high interest rates). While I believe such policies would work again, I wonder how we could afford them today, given the total debt in the U.S. And that's a big problem.
We might not be able to stop a run on the dollar, simply because we're broke.