Kahn-Markovic show that every hyperbolic 3-fold contains an immersed $\pi_1$ injective surface. Are there any known examples of hyperbolic 3-folds that do not contain a embedded $\pi_1$ injective surface?
Infinitely many Dehn fillings on the figure eight knot complement $M_8$ have this property:
Take a hyperbolic filling on a non-boundary slope. If this were Haken, then, by a cut and paste argument, any incompressible surface $F$ could be made to intersect $M_8$ in an incompressible boundary incompressible surface, contradicting the choice of slope.