Let's consider rotating charged ring. Theoretically mass of this ring has no limit as rotation speed increases. So what about magnetic moment of the ring? Is it limited by the value of speed of light?
The magnetic moment of a rotating charged ring is $$ m=IA=\lambda v A. $$ Here $I$ is current, $\lambda$ is linear charge density, and $A$ is the enclosed area. This expression is true even relativistically. The quantities $\lambda,A$ don't depend on the rotation speed, so the magnetic moment is limited to $\lambda c A$.