Consider the function $f:z\mapsto 1/\sqrt{z}$, defined, say on the right half-plane $Re(z)>0$. (We can resolve ambiguity by taking the branch that is positive for real $z$).
Let $U$ be the right half-plane, and $a=0$.
Then the following conditions, lifted from Wikipedia's page on essential singularities, appear to hold:
1) $f(z)$ is not defined at $a$ but is analytic in the region $U$. Moreover, every open neighborhood of $a$ has non-empty intersection with $U$.
2) $\lim_{z\rightarrow a}f(z)$ does not exist.
3) $\lim_{z\rightarrow a}{1\over f(z)}$ exists (and is equal to zero).
According to that Wikipedia page, it follows that $a=0$ is a pole of $f$.
But it seems to me that $f$ has no Laurent series at $a=0$, which makes me skeptical that this really is a pole.
This seems to leave three possibilities: Either Wikipedia is wrong, or I am wrong, or I have misunderstood Wikipedia. Which of these is correct?
The conditions in the "Alternate descriptions" section of the Wikipedia article are wrong. The correct definition of a pole (as the term is normally used in complex analysis) requires that $f$ be defined on an entire deleted neighborhood of $a$, not just in an open set that intersects every neighborhood of $a$ (that is, a pole is by definition an isolated singularity). Since the right half-plane does not contain any deleted neighborhood of $0$, your function $f$ does not have have a pole at $0$.