What do you call a person who incessantly thinks only they are good enough to do something?

Friends may half-jokingly call that person the little dictator/tyrant. This is slang but good.

However I think the best word for your details is imperious.

assuming power or authority without justification; arrogant and domineering

Some other words to describe this person would be:

• overbearing... unpleasantly or arrogantly domineering
• to be "nice" you can say assertive... having or showing a confident and forceful personality
• usually used for a woman (by other women) would be haughty... arrogantly superior and disdainful.
• and the common man's term would be simply pushy... excessively or unpleasantly self-assertive or ambitious.

You might call them perfectionists since they feel only their efforts are "good enough."

Given the level of self-involvement described, I am leaning toward narcissist.

I like egomaniac for this one too.

know-it-all 1. a person who acts as though he or she knows everything and who dismisses the opinions, comments, or suggestions of others.

Consider peremptory.

peremptory: characterized by often imperious or arrogant self-assurance.