The [Turing degrees](http://en.wikipedia.org/wiki/Turing_degree) are an immensely intricate poset $\mathcal{D}$.  Here are some of their remarkable properites:

1. Every countable poset is embeddable in $\mathcal{D}$.
2. $\mathcal{D}$ contains *minimal* degrees.  (a non-zero degree $\mathbf{m}$ with no degree between $\mathbf{0}$ and $\mathbf{m}$)
3. For every degree $\mathbf{d}$, there is a degree that is incomparible with $\mathbf{d}$. 
4. $\mathcal{D}$ contains an antichain of size $2^{\aleph_0}$.
5. No infinite strictly increasing chain in $\mathcal{D}$ has a least upper bound.  
6. For every degree $\mathbf{d}$, there is a degree $\mathbf{c} < \mathbf{d}$ such that $\mathbf{c}'=\mathbf{d}$.  (here $\mathbf{c}'$ denotes the *jump* of $\mathbf{c}$)
7. For any two recursively enumerable degrees, there is a recursively enumerable degree strictly between them.  
8. Any finite distributive lattice can be embedded in the recursively enumerable degrees.