  We investigate the structure of finite sets $A \subseteq \Z$ where $|A+A|$ is
large. We present a combinatorial construction that serves as a counterexample
to natural conjectures in the pursuit of an "anti-Freiman" theory in additive
combinatorics. In particular, we answer a question along these lines posed by
O'Bryant. Our construction also answers several questions about the nature of
finite unions of $B_2[g]$ and $B^\circ_2[g]$ sets, and enables us to construct
a $\Lambda(4)$ set which does not contain large $B_2[g]$ or $B^\circ_2[g]$
sets.
