Question

When it comes to entropy, how are we able to get a single maximum entropy for large systems?

Answer 1

In statistics and information theory, a maximum entropy probability distribution ... In connection with maximum entropy distributions, this is the only one needed,.. ☺ ☺

Answer 2

Entropy is maximum at spontaneous process than at equilibrium. According to the second law of thermodynamics, "increase in entropy always tends to be spontaneous". so a process which having maximum entropy will be a spontaneous process.some thermodynamics books said about state of entropy as,
          ∆E>0-spontaneous
          ∆E<0-non-spontaneous
          ∆E=0-equilibrium



Maximum entropy mean everything is as randomized as it can be.
This exactly what you expect from equlibrium, everything is spread evenly (randomized). 

When all gas molecule occupy just half the room, this cannot be equlibriun since molecules can cross to the other side.
When molecules are everywhere randomly the rate of molecules miving from left to right is equal to the opposite flow (on averagre) and we get equilibrium.