What is this word?
If we're accustomed to the biblical reading we'd know it appears several times in the bible: 2Ki 23:22; Psa 94:13 and others.
If you look for it at your dictionary you will hardly find it.
It comes from the basic noun יוֹם, day.
The first מ is the well known preposition מִן, from, since. It has lost its final nun and has been prefixed to the following word.
The remaining ימי is the plural construct of יוֹם, so we have מִימֵי = from/since (the) days of.