According to Official Guinness Records,
The most stairs climbed on a unicycle is 1,005 and was achieved by Owen Farmer (USA) in San Francisco, California, USA, on 3 April 2016.It took Owen 46 minutes and 50 seconds to climb all 1,005 stairs.
For a complete list of 2016 records, please visit 2016 Guinness Records in United States.