Let $a,$ $b,$ and $c$ be positive real numbers such that $a^2 = bc$ and $a + b + c = abc.$  Find the smallest possible value of $a^2.$
By AM-GM,
\[abc = a + b + c \ge 3 \sqrt[3]{abc},\]so $(abc)^3 \ge 27abc,$ which means $(abc)^2 \ge 27.$

Since $bc = a^2,$ $a^6 \ge 27,$ so $a^2 \ge 3.$

Equality occurs when $a = b = c = \sqrt{3},$ so the smallest possible value of $a^2$ is $\boxed{3}.$