Someone who thinks they are overly special/out of the ordinary

A special snowflake is someone who feels he is different (read: better) than everyone else.

This term is quite informal and has gained a lot of popularity recently; I often see it used in criticism against social justice warriors. Its origin lies on the oft-repeated trivia that no two snowflakes are the same.

Megalomaniac may apply to your description:

• A megalomaniac is a pathological egotist, that is, someone with a psychological disorder with symptoms like delusions of grandeur and an obsession with power. We also use the word megalomaniac more informally for people who behave as if they're convinced of their absolute power and greatness. Your garden-variety egomaniac might be self-centered and overconfident but he's a little softy compared with the megalomaniac who wants to control the world.

Vocabulary.com

Hubris would be the word.

[mass noun] Excessive pride or self-confidence.
‘the self-assured hubris among economists was shaken in the late 1980s’

(oxforddictionaries.com)