I am looking for a word which can be used to describe someone who is very "comfortable in their skin", secure with themselves, and most importantly completely immune to being embarrassed or mocked.
Some examples might include consistently mispronouncing a word in a conversation (like "pirate" instead of "pilot") without feeling silly, even though others try to mock him/her. Or more seriously, not feeling embarrassed at all if their private photographs (ie in the shower or some other scandal-like situation) were leaked.
Words like "unashamed", "unabashed" or "unapologetic" usually have a negative connotation and do not seem applicable, since the person is not necessarily doing things which society would consider shameful from a moral standpoint.
Best Answer
unflappable
I like this word better than imperturbable for the idea of being cool and collected, even in the face of intentional attempts to embarrass or harass.