Okay, I may be overthinking this, but I don't understand how multiple turns of maenuver acceleration make the ship go faster than the M rating. If an M1 ship accelerates to max, it's going M1. That's the level of its acceleration; there's no more energy to push out or push against. So a second "turn" of M1 would see it going M1. If you're flying a plane that goes 100 MPH, after three hours, it's still just going 100 MPH. A bigger engine can take a different plane to a different speed, but it still doesn't accelerate past its max.
What am I missing?
Newton's laws of movement.
Law 2
force=mass x acceleration or F=ma
Law1
an object continues in its state of motion unless acted upon by a (resultant) force
a bit of further explanation here, an object often has more than one force acting on them, the resultant force is the overall force that is the result of considering these forces.
So a spaceship is in orbit. For the sake of simplicity we will ignore its circular movement around the planet and assume it has just assumed escape velocity.
It fires its 1g drive, or rather its 1g drive provides a force that accelerates the mass of the ship to 9.8m/s in 1 second.
If the drive shuts off the ship continues on its merry way with a velocity of 9.8m/s - for ever
If the drive continues to apply the same force the ship continues to accelerate, so after 2 seconds the ship now has a velocity of 19.6m/s
For the magic reactionless m-drive provided the maneuver drive can be supplied with energy then the drive will continue to apply a force which will continue to accelerate the ship, once you start to hit relativistic speeds things change.
For a reaction engine there is a top speed which is limited to the speed at which you can throw your exhaust products out the back of an engine (this is a gross simplification).
So can an m-drive get close to c?
According to CT, it's up to you, but there are examples of ships travelling at fractions of c all the way up to 0.9c in the boardgames.
Other iterations of the games have different physics - for example in T5 and MgT the m-drive is only efficient out to 1000D, the TNE HEPlaR drive has to obey the rocket equation so couldn't carry enough fuel to get anywhere near c, that sort of thing.