The topological obstructions on the attitude space of a rigid body make global asymptotic stabilization impossible using continuous state-feedback. This paper presents novel algorithms to overcome such topological limitations and achieve arbitrary attitude maneuvers with only continuous, memory-less state-feedback. We first present nonlinear control laws using both rotation matrices and quaternions that give rise to one almost globally asymptotically stabilizable equilibrium along with a nowhere dense set of unstable equilibria. The unstable equilibria are uniquely identified in the attitude error space. Pseudo-targets are then designed to make the controller believe that the attitude error is within the region of attraction of the stable equilibrium. Further, the pseudo-target ensures that maximum control action is provided to push the closed-loop system toward the stable equilibrium. The proposed algorithms are validated using both numerical simulations and experiments to show their simplicity and effectiveness.