Review: A doubly-periodic meromorphic function $f$, or an elliptic function, has a degree $d$ defined to be the sum of the orders of the poles in its fundamental parallelogram $\Gamma$. As $z$ ranges over the $\Gamma$, $f(z)$ winds over the Riemann sphere $d$ times. Elliptic functions must have degree $d>1$.
For physics reasons, I would really like to have an "almost elliptic" function of degree $d=1$. I can't give up periodicity, but I can give up strict meromorphicity everywhere.
Is there such a function which is meromorphic outside a measure zero set? The measure zero set could be $\partial \Gamma$ for instance. Or can you prove you can't have it?
More generally, can you provide an "almost" elliptic function with degree $d=1$ for a suitable interpretation of "almost"?