Used adverbially to mean 'whenever something happens' or 'on every occasion'. Often followed by a verb phrase describing a repeated action or state.
彼は会うまいじ、新しい話をしてくれる。
Every time I see him, he tells me a new story.
まいじのことだが、彼は遅刻する。
As is the case every time, he is late.
Compound of 毎 (every) and 次 (next, order). The exact historical derivation is uncertain, but it follows a pattern of Sino-Japanese compounds expressing repetition.