Trip to Space -- Can ship with 1g acceleration escape Earth?

[Nugatory]

In the game, 1 hex is equal to the distance traveled at 1g acceleration in the course of a turn.

That makes no sense, unless the one turn of the game is supposed to represent a different amount of time for different players and different stages of the game, and I doubt you intended that. The problem is that the distance traveled depends on the initial velocity as well as the acceleration, so the same 1g acceleration will mean different distances traveled and "the distance traveled at 1g acceleration" is ill-defined. The original 1973-vintage Triplanetary game doesn't make this mistake (and it may be that you just meant to say something different here than what you actually said). The original game also does allow the ship to escape earth, because of the rule that vectors that originate in a gravity hex are not affected by the hex.

However... Although that early version of the game did make a creditable effort (given the limitations of the grease-pen mechanics - these days we could play it on a whiteboard with a hex grid projected on it) to model the effects of gravity and acceleration, it really has very little to do with any real physics. We help you with understanding what physics really says, but that's no longer what this discusion is about, so it is closed.