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When I come across the word maxime in macronized texts, it usually lacks a macron over the first vowel. In Ørberg's Lingua Latīna series, however, in which the macrons are (from what I understand) very reliable, it's always macronized māximē. At first I thought that perhaps the macron was there to mark a vowel made long by position before the letter "x," but in other words with "x" (exercitus, mox, vexāvit, and so on) there's no macron.
So what's the deal? Is the vowel long by nature or not?
On the one hand, we have MAX(IMO), A with an apex in CIL VI 2080 17
However, as De Angelis and Chilà 2015 put it,
"the interpretation of the vowel of maximus as long is anything but certain" (p. 92).
Allen thinks the long a is doubtful there (Allen 1978: 70). Forston argues the apex there could be a mark of the "final member of the phrase" (Weiss 209: 175) or it could be a scribal error. Kortland finds this evidence "too weak for any conclusions."
"The long a camp": Leumann 1977 gives māx- and mentions it in §120 (Lachmann's Law); so does Tronskii 1960 (p. 99). Jasanoff 2004 also puts it together with two other non-participles under same Lachmann's Law (p. 414).
Also see http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0060%3Aentry%3Dmaximus1
Weiss takes a cautious position on this - he mentions it in the section devoted to Lachmann's Law, with a caveat.
I've never seen a macron'd ā in maximus before. But the evidence is sparse on this matter; since the first syllable is always long by position, poetic meter doesn't tell us anything about the vowel quantity.
Maximus comes from the root of magis, which is known to have a short a, and mājor, which had a heavy first syllable but was probably a short a and long jj (see the note in this answer). So etymologically it seems to be short. But vowel quantities can change when the sounds around them change, so this isn't foolproof evidence.
