Being so absorbed in something that one forgets oneself; completely engrossed (yojijukugo).
彼は研究に無我夢中で取り組んでいる。
He is working on his research with complete absorption.
子どもたちは遊びに無我夢中だった。
The children were totally absorbed in playing.
無我夢中で歌っていると時間を忘れる。
When I'm completely lost in singing, I lose track of time.