The standard, medium sized edition is refered to as 42.2 (5 teams). This takes about 4-6 hours to play in a competitive game, and costs 45/70 (amzn vs retail). This is the game most often played in tournaments.
I call that version “1985”, the backronym for it is 1942. The old edition is still played at Gencon, but the 42.2 game is the same scale, with some of the updated pieces and rules. As you may suspect, this 1985 edition is playable but feels outdated because defending yourself at sea is a bit difficult because of the piece selection and smaller economies.
A “Cadillac” version of the game was also created, this is two separate boxes that are refered to as “Global 1940”. These sets are (70/90 x2) more expensive, but you get the full flavor of the rules and the sweeping scale of the whole world. This game is the gold standard for extremely dedicated play (8-10 hours or more, 7 teams).
There are also custom games such as Global War 1936 that are spin-offs of AxA. These games are more complex and expensive, but they are more fully “strategic” games. They are not printed by WoTC.
There are also 5-6 obsolete versions of AxA, which have their own maps and layouts, but are no longer in print and use a less modern ruleset. There is a 50th edition set, which is the rarest as it was a collectors edition and the last copy I saw was $300+, though Global is far better and similar.
Many of these games are considered solid but flawed, such that one team or the other (usually the Axis) is considered to have a decisive advantage with the out of box rules. Adding or removing select pieces can address these concerns.