Here are two examples  suggesting the complexity of the  world of $3$-manifolds.

The first is the classical result  that any  $3$-manifold   can be obtained by integral surgery on a link in $S^3$.  If you believe that knots and links  form a complex Universe, than this result should suggest that $3$-manifolds    cannot be much simpler.

The next example comes from the striking work of  [Dunfield and Thurston][1]  on random $3$-manifolds.   You can get such things by picking  random elements  in the mapping class  group, where randomness is generated by a random walk on this group  This has lead to the discovery  of  strange $3$-manifolds. For more recent  work on this topic see also this paper of [Lubotzky, Maher and Wu][2].


  [1]: http://arxiv.org/pdf/math/0502567.pdf
  [2]: http://arxiv.org/pdf/1405.6410.pdf