So... this is one of those fun areas that seems to get overlooked a lot.
The ISS is experiencing a constant gravitic acceleration of a little less than 1G (relative to Earth), but the mechanical stresses are negligible because its sitting in a near-uniform gravitic field. If we were to move it to, say, Jupiter, that number would likely increase (depending on the oribtal distance), but the effect on the station (and its occupants) — at least due to the increased acceleration — would be negligible.
This is because of the different way that gravity and conventional propulsion systems work. A rocket (or ion thruster, Orion drive, ...) works by transmitting a force to some object (pusher plate, back wall of the rocket nozzle, etc.). That force must then be translated mechanically through the structure of the space ship and, if it's manned, the bodies of its crew. This is also why you "feel" acceleration. Take standing on a planet; gravity is pulling on you uniformly, but the ground/floor/whatever is opposing that force. However, that opposing force is only being applied to a small part of you (e.g. the bottoms of your feet). That force then gets transmitted through your bones and tissues. In water, you feel lighter because this force is much more spread out, while in free fall the opposing force (nearly) goes away, even though you are still accelerating.
What does this mean for our hypothetical drive?
If the drive produces a uniform gravitic field, I can't think of any reason why there should be a mechanical limit; the limits will be "whatever the drive can do given how much power you can feed it" (which we're ignoring).
That said, a uniform gravitic field is probably not plausible, since AFAIK such a thing does not exist in nature. Rather, gravity (at any point) is:
$a_g = \sum \frac {GM_pV_p}{|V_p|^3}$ for all points of matter, where:
- $G$ is the gravitational constant
- $M_p$ is the mass of each such point
- $V_p$ is the direction vector from wherever we are measuring gravity to such point
Since far-away masses have near-zero influence and close-together masses act almost like a single mass, we can usually simplify this (also ignoring direction) to:
$a_g = \frac {GM}{d^2}$
Let's say that, rather than producing a uniform field, our hypothetical drive produces a point of immense "virtual mass". Now our drive looks like falling into a gravity well, except that the center of gravity conveniently keeps receding such that we never reach it. (Again, we're ignoring the pretzel this makes out of physics as part and parcel of the whole idea of "artificial gravity".) Now we do have a practical limit, because different parts of the ship are subject to different gravitic fields. This difference is "shear" or "tidal force", and too much of it isn't good for ships (or people). At sufficient levels, this leads to the delightfully-named effect of spaghettification.
This is why you hope your drive really can create a uniform field, or at least, can create multiple and/or spread out "virtual masses" in a way that is carefully tuned to minimize shear within the ship's volume. (Shear outside the ship can be tremendously useful as a defense, since it may be nigh-impenetrable, potentially even to photons.)
Suffice to say, the mathematics for computing maximum gravitic shear can get complicated. I'm also unsure how much shear the average human can take, though I wouldn't be surprised if 1G is structurally acceptable. (The effects it would have on equilibrium may be another matter! On the other hand, How much variation in gravity between feet and head is noticable? suggests I might be wildly optimistic with that number.) Ironically, a large spaceship might actually be more susceptible to shear than its crew.
