A system is said to be at equilibrium, if the entropy of the system has reached __________ value.

According to second law of thermodynamics, the entropy \[{\left( {dS} \right)_{universe}} > 0,\]    for an irreversible or spontaneous process and \[d{S_{universe}} = 0\]   for a reversible process or almost equilibrium reached process. so for an natural process since there will be irreversibility always the system entropy goes on increasing until equilibrium is reached where the system entropy becomes equal to zero. Hence a maximum exists for a system.