Two states have no Daylight Saving Time: Arizona and Hawaii.
Arizona tried DST for one year, back in 1967, but in 1968 state legislators decided it wasn’t right for them, and so they abolished it. A bill was introduced this year (2015) to consider adopting it again, but that bill was withdrawn in less than a week due to opposition from Arizona residents. A small pocket of Daylight Saving Time exists on the Navajo nation located inside the state boundaries, but they control their own laws and regulations, and are not bound by state statutes.
Hawaii also does not observe Daylight Saving Time.