According to this Wikipedia list, which comes from the CIA World Factbook, the only country with no debt at all is Brunei.
However, I think that’s misleading. There are any number of countries–Saudi Arabia, Kuwait, Norway, and China come to mind–that have sovereign wealth funds equal to or greater than their government debt. These are countries that are either (a) resource intensive; or (b) where the state owns a substantial share of the economy; or (c) both.
I don’t have time to merge the two lists, and wouldn’t trust the results if I did, but I believe you would find a number of countries with negative “net government debt” if sovereign wealth funds were subtracted out.
This is not a country but the Canadian province of Alberta recently paid off its debt, mostly thanks to oil and gas revenues. The government has plans to start borrowing again though.