Total Absorption
Losing Oneself In, Frantic
noun, の adjective
You already know that 無我 is "selflessness" and 夢中 is "crazy about." So this means something like total absorption. You're so into something that you’ve started losing your sense of self. Just like 夢中, this can also describe doing something frantically, like running away from somebody or something.
This is a jukugo word, which usually means on'yomi readings from the kanji. If you know the readings of your kanji you'll know how to read this as well.
ゲームに熱中しすぎて、無我夢中の状態になっていた。
I got so absorbed into a game, that I was completely losing myself in it.
「撃て！撃つんだ！」とビエトは無我夢中でさけんだ。
"Shoot! Fire!" Viet shouted frantically.
正直、無我夢中だったので、何も覚えていません。
To be honest, I don't remember anything because I was completely absorbed in it.
彼は無我夢中で働き、二十代で家を建てたが、三十代で墓を建てることとなった。
Working feverishly, he built his house in his twenties and built his tomb in his thirties.