Incorrigible

[Sarah]

A person who believes that they are always right, and can NOT be corrected, or will not accept that they are wrong about anything

[Justin Bieber]

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.

[Little_Girl]

Hopelessly stubborn; to the point where your mother calls them incorrigible.