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 $ ).