To not care at all / not even bother to care (emphatic).
彼女は周囲の噂を気にもしないで自分の道を進んでいる。
She doesn't even care about the surrounding rumors and follows her own path.
子供たちは人の目を気にもしないで外で遊んでいた。
The children were playing outside without caring what people thought.
上司の注意を彼は気にもしなかった。
He didn't even pay attention to the boss's warning.