Let:
$$ f(z) = \sum_{j=1}^{\infty} \frac{1}{(z - j)^{j}} $$
so that $f$ has a pole of order $j$ at $z = j$ for each integer $j > 0$. Is $f$ meromorphic in the extended complex plane? It is analytic except on a discrete set of poles, so I think the answer is "yes," but $f$ isn't a rational function, and I know that every function that is meromorphic on the extended complex plane is rational.
Am I misunderstanding the definitions here?