incorrigible

Short definition from Wiktionary

Adjective

1. 1
   Defective and impossible to materially correct or set aright.
2. 2
   Unmanageable; impervious to correction by punishment or pain.
3. 3
   Incurably depraved; not reformable.
4. 4
   Unchangeably established in a belief or habit.
5. 5
   Intrinsically incapable of being corrected; impossible to disprove, by its very nature.
6. 6
   Impossible to cure.

Noun

1. 1
   An incorrigibly bad individual.