Here's a simple birth model which leads to power law behaviour with exponent 1.
Start with a single individual of type 1.
Reproduction as follows:
(a) each individual produces "clone offspring" (a child of the same type as itself) at rate 1.
(b) in addition, each individual of type 1 produces "mutant offspring" (a child of a new type not yet seen before) at rate $\mu$, where $\mu$ is any positive constant. So the first mutant will be called type 2, the second type 3, etc.
Let $N_k(t)$ be the number of individuals of type $k$ alive at time $t$.
Once the first individual of type $k$ has been born, the type-$k$ family grows exponentially. Also, the first individual of type $k$ is born at time $\log k + O(1)$.
From this it's quite easy to obtain that $N_k(t)$ behaves something like $e^t/k$. More precisely, for any $k$ the quantity $ke^{-t}N_k(t)$ converges as $n\to\infty$ with probability 1 to some random variable $W_k$, say, and the sequence of distributions of $W_k, k\geq 1$ is tight.
Reordering the $N_k(t)$ into decreasing order still leaves essentially the same rate of decay.
So for large $t$, $N_k(t), k \geq 1$ obeys Zipf's law (for a range of $k$ that depends suitably on $t$, say $ k < < e^t $ ).