Literally: nothing – self – dream – middle / inside
Alternately: Utterly absorbed in something; a state of deep concentration or focus. This is usually positive, but it can be negative, as in losing oneself in something; losing control and failing to pay attention to the rest of the world.
Notes: Apparently some people carelessly write 夢我 for muga or 無中 for muchuu; naturally, these are errors. Replacing 夢中 with 無心 (mushin), a Buddhist term for being “free from obstructive thoughts,” though, produces a synonymous phrase. There are a number of other synonyms as well, including 一心不乱 (isshin furan), literally “single-minded, without disorder.”
This is a compound of compounds. 夢中 (“entranced,” “dreaming”) may be encountered on its own somewhat commonly, but 無我 (“self-effacement”) is more obscure and more strongly connected to a Buddhist idea that even individual souls do not truly exist.
It’s a bit of a surprise that I hadn’t done this compound yet; it’s one of the more well-known yojijukugo, and all the characters have formally been taught by the end of elementary school..