![alt text][1x] [(source)][1]

The exterior of Suzuki's Brunnian graph on $n$-edges, here pictured with $n=7$, is irreducible, atoroidal, boundary incompressible, and acylindrical.  See 

Luisa Paoluzzi and Bruno Zimmermann. On a class of hyperbolic 3-manifolds and groups with one defining relation. _Geom. Dedicata_, 60(2):113–123, 1996 

or  

Akira Ushijima. The canonical decompositions of some family of compact orientable hyperbolic 3-manifolds with totally geodesic boundary. _Geom. Dedicata_, 78(1):21–47, 1999.

(I think these manifolds may be contained in Bruno's list also.)

  [1]: http://www.math.wisc.edu/~kent/Suzuki.jpg
  [1x]: https://i.sstatic.net/cfBtsm.jpg