CattaraugusThrills.com . . . Cattaraugus, NY: founded in 1806...Population as of 2000: 1,075. The county was formed from Genesee... Ellicottville, located south center of the county, was chosen as the shiretown.... Cattaraugus didn't have a County Government untill 1817... Its records were first kept in Buffalo, but destroyed in 1817 when the British burned the County Courthouse.