The folks at Planet Money have dug up a remarkable document: a 2000 Clinton administration document examining the risks that would be created by paying off the federal debt. As hard as it is to imagine now, that was a real prospect at the time.
Although the report never saw the light of day (Planet Money obtained it through a Freedom of Information Act request), it outlined the possible harmful consequences of retiring the debt completely. The problem is that if the debt were paid off, there would be no more U.S. Treasury bonds left in circulation. Treasury securities are crucially important to the world financial system in a number of ways: banks buy them as low-risk assets, the Fed uses them for executing monetary policy, and mortgage interest rates vary based on Treasury rates.
In retrospect, that was a good problem to have.