A person who believes that they are always right, and can NOT be corrected, or will not accept that they are wrong about anything
What a know-it-all that guy is incorrigible
someone who can not be changed or modified. They can not be MADE better, they must be themselves. Often associated with people who are also considered precocious or endearing. They march to the beat of their own drum and are very rooted in their beliefs and can not be swayed by others who have different beliefs.
Sometimes considered a smart-ass or trouble maker or irritating.
Peter's friend called him incorrigible after he was texting one word-er onomatopoeia, instead of answering the questions he was asked.