A display or example of someone being a pompous ass. Typically refers to those who are arrogant and believe they are better than you, and act accordingly.
A person behaving with pompous-assery is acting even more pompous, arrogant, pretentious, and douchey than usual, or is behaving as such out of character.
His obnoxious arrogance that day was such a display of pompous-assery I could barely contain myself.