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 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.