We had "nearly" paid of the national debt in 1980.  It was just below $800 billion at that point, after Truman began paying off WWII and the Great Depression:

    Reagan: 1980 was 907B   - added $2.1T, up 275% in 1988 to $3T
    Bush I: 1988 was $3T    - added $1.8T, up 63%  in 1992 to $4.8T
    Clinton: 1992 was $4.8T - added $1.2T, up 22%  in 2000 to $6T
    Bush II: 2000 was $5.1T - added $6.6T, up 129% in 2008 to $11.7T
    Obama: 2008 was $11.7T  - added $4.3T, up 37%  in 2012 to $16T

Compiled directly from the numbers at the Dept. of the Treasury: http://www.treasurydirect.gov/govt/reports/pd/histdebt/histdebt.htm

But to answer the question directly, I guess the *last* President to pay off/reduce debt was Carter, though all of them since WWII had been doing it.

**EDIT** - With consideration to GDP, looking at spending vs revenue, a picture paints a thousand words.  This is where each president left the Debt, as a percentage of what "they had to spend". ![**National Debt as a Percentage of GDP**][1]


  [1]: https://i.sstatic.net/MrQyR.png