There are a lot of assumptions here, such as Room Temperatur Superconductors and unlimited energy. But ok, lets go with it.
Limitations
Assuming a K2 Civilisation, the only real limitation is Physics itself. For everything that follows, ill assume Earth as the launch Platform.
Earth to the Stars
Earth has a circumference of 40.075km . So or Maximum Rail (Assuming no spiral pattern) is 40.075km long. The Human body is fit for a Horizontal Maximum acceleration of around 6g for 10min. But that is only for 10min. So lets assume the normal, everyday Human can Handel 2G for unlimited time.
This gives us a Maximum Acceleration of 2G (=19.62m/s²) for a Track lenght of 40.075km. We can use the equation v² = u²+2as. In which v is or final velocity, u² is or inital velocity, a is or acceleration and s is or track lenght. All in SI Units.
So v² = u²+2as
v² = (0m/s)²+2*(19.62m/s²*400750000m)
v² = 15725430000m/s
v = sqrt(15725430000m/s)
v = 125401.07655m/s /:1000
v = 125.40km/s
So we can expect a final velocity of around 125.4km/s with such a Track and only one revolution.
The time for this one revolution of course decresses the faster we go. We can find the time t with the equation v = u+a*t. In which v is the final velocity, u is the inital velocity, a is the acceleration and t is the time.
v = u+a*t /-u
v-u = a*t /:a
(v-u:a) = t
v/a = t
125401.07655m/s / 19.62m/s² = t
t = 6391.4921789s / 60
t = 106.524869648min / 60
t = 1.77541449414hr
So it would take around 1.8hr for the first revolution.
But, i hear you ask, why is any of that important ? Well, because as you can see, one revolution really dosnt give us much in terms of speed. But, i am about to prove that wrong.
Gravity Time
The earth has an acceleration of 9.81m/s². And there is this force, you probably know it, the centripetal force. If you think about it, this Mass Driver is nothing else then a giant centripetal force generator.
So how great would this force be with a Velocity of 125.4km/s ?
Firstly, i know what you are thinking, "but dosnt the centripetal force point inwards ?" And yes it does but also, it dosnt matter since the outwards pointing vector is the same. Magic.
The equation to get the force is just F_z = m*v² / r. In which, oh god we have a problem. We need mass dont we ? And thats where you are wrong.
We just look for the centripetal acceleration. For which the mass is not needed. Because:
F = m*a
F_z = F
ma = mv² / r -> m cancels out
a = v² / r
a = (125401.07655m/s)² / 6371000 m
a = 2468.28284412 m/s
a = 2.4km/s
Yeah thats right, when you shoot along a Rail at 125km/s you get pushed upwards with an velocity of 2.4km/s Which for all inteneds and pruposses, is equal to an acceleration force because the Train is still moving. Meaning those 2.4km/s are your effectiv Gravity.
But we can do one better. Lets assume a Train without mass and only a 80kg Human riding along. In this case F_z = m*v² / r is true.
Fz = m*v² / r
Fz = 80kg*(125401.07655m/s)² / 6371000 m
Fz = 197462.62753 N
And as F = m*a, we can solve for a by doing the old swiggity swoody
F = m*a -> /a
f/m = a
197462.62753 N / 80kg = a (we get acceleration because N = kg*m/s²)
a = 2468.28275 m/s²
a = 2.4km/s²
So yeah, this should come as no supprise. The Momentary Velocity is the same as the General Acceleration. And as we know, Mass dosnt play into effect when it comes to Acceleration (in simple terms).
The Problems
So i think you can start to see the problem. In theory, a Rail along the Planet could push stuff put to very high speeds. But, the centripetal fore / acceleration really dosnt see it that way. Most electronics would die with an Acceleration of 2.4km/s². Not to mention that no human can get out of there alive.
El Answer
But your question was who fast this could get. Well, going back to the question how many G´s a Human can handel, its around 2G. All we need to do is solve for lets say 20m/s² to make it simple.
We can use a = v² / r and just solve for v². As this is or final velocity.
a = v² / r
a*r = v²
sqrt(a*r) = v
v = sqrt(20m/s)*6371000 m))
v = 11288.0467752/s
v = 11.288km/s
So a final velocity for such a system that is intended for humans is around 11.3km/s. Which is really fast.
Assuming normal Cargo can sustain around 100G´s we have a Velocity of:
a = v² / r
a*r = v²
sqrt(a*r) = v
sprt(1009.81m/s²6371000 m) = v
v = 79056.6316004 m/s
v = 79km/s
Ti Solve
Now, this all seems like a bit of BS. You got this long Track and the best we can pull of are 80km/s ?
But really, the only reason why we are limited is because of the shape. If it was a line track with no curve, you could go as fast as you want. There is no Upwards force.
Not to mention that i would like to see a Rail Track hanging on earth with 100times the Gravity pulling on it once your super Train goes over it.
All in all, its a very unsmart idea to build something like that.
Sorry for typos, i am German D:
