before vowels mechan-, word-forming element meaning "pertaining to mechanics or  mechanisms; done by machine," from Latinized form of Greek mekhano-, combining form of mēkhanē "device, tool, machine; contrivance, cunning" (see machine (n.)).

word-forming element meaning "of or pertaining to medical science; from a medical standpoint; pertaining to medicine and," used as a combining form of Latin medicus "physician; healing" (from PIE root *med- "take appropriate measures").

word-forming element meaning "middle," used as a combining form of Latin medius "in the middle, between; from the middle," from PIE root *medhyo- "middle."

before vowels meg-, word-forming element often meaning "large, great," but in physics a precise measurement to denote the unit taken a million times (megaton, megawatt, etc.), from Greek megas "great, large, vast, big, high, tall; mighty, important" (fem. megale), from PIE root *meg- "great." Mega began to be used alone as an adjective by 1982.

High-speed computer stores 2.5 megabits [headline in "Electronics" magazine, Oct. 1, 1957]

word-forming element meaning "black," from Greek melano-, combining form of melas (genitive melanos) "black, dark, murky,"probably from a PIE root *melh-"black, of darkish color" (source also of Sanskrit malinah "dirty, stained, black," Lithuanian mėlynas "blue," Latin mulleus "reddish"). 

before vowels mer-, word-forming element meaning "part, partial, fraction," from Greek meros "a part, a fraction," from PIE root *(s)mer- (2) "to get a share of something."

word-forming element of Greek origin meaning 1. "after, behind; among, between," 2. "changed, altered," 3. "higher, beyond;" from Greek meta (prep.) "in the midst of; in common with; by means of; between; in pursuit or quest of; after, next after, behind," in compounds most often meaning "change" of place, condition, etc. This is from PIE *me- "in the middle" (source also of German mit, Gothic miþ, Old English mið "with, together with, among"). Notion of "changing places with" probably led to senses "change of place, order, or nature," which was a principal meaning of the Greek word when used as a prefix (but also denoting "community, participation; in common with; pursuing").

The third sense, "higher than, transcending, overarching, dealing with the most fundamental matters of," is due to misinterpretation of metaphysics (q.v.) as "science of that which transcends the physical." This has led to a prodigious erroneous extension in modern usage, with meta- affixed to the names of other sciences and disciplines, especially in the academic jargon of literary criticism: Metalanguage (1936) "a language which supplies terms for the analysis of an 'object' language;" metalinguistics (by 1949); metahistory (1957), metacommunication, etc.

word-forming element meaning "small in size or extent, microscopic; magnifying;" in science indicating a unit one millionth of the unit it is prefixed to; from Latinized form of mikros, Attic form of Greek smikros "small, little, petty, trivial, slight," perhaps from PIE *smika, from root *smik- "small" (source also of Old High German smahi "littleness"), but Beekes thinks it a Pre-Greek word.