。
There is no woman as arrogant as she is.
Literal
She [to the extent of-ほど] arrogant woman [topic-は] does-not-exist.
The ~ほど~ない pattern expresses 'nothing is as X as Y.' The literal structure is 'a woman who is arrogant to the extent of her does not exist.' 傲慢 (ごうまん) is a Sino-Japanese compound meaning 'arrogance, haughtiness' — somewhat advanced vocabulary with low-frequency kanji.