A new code for computing fully general relativistic solutions of strongly magnetized rapidly rotating compact stars is developed as a part of the Compact Object CALculator (cocal) code. The full set of Einstein's equations, Maxwell's equations, and magnetohydrodynamic equations are consistently solved assuming perfect conductivity, stationarity, and axisymmetry, and strongly magnetized solutions associated with mixed poloidal and toroidal components of magnetic fields are successfully obtained in generic (noncircular) spacetimes. We introduce the formulation of the problem and the numerical method in detail, then present examples of extremely magnetized compact star solutions and their convergence tests. It is found that, in extremely magnetized stars, the stellar matter can be expelled from the region of strongest toroidal fields. Hence, we conjecture that a toroidal electrovacuum region may appear inside of the extremely magnetized compact stars, which may seem like the neutron star becoming the strongest toroidal solenoid coil in the Universe.