This question arises in the context of a question asked on MSE: Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$. Part of the answer to that question is the question above which is being asked here since it appears to be at a "research level".

Is there an example (or even an obstruction) to embedding an open connected Riemann surface into $\mathbb{C}^2$?

A theorem of Narasimhan (Narasimhan, R. "Imbedding of open Riemann surfaces", Göttingen Nachrichten, No. 7 (1960), pp. 159-165; also see American Journal of Mathematics Vol. 82, No. 4 (Oct., 1960), pp. 917-934) proves that any open connected Riemann surface has a non-singular embedding in $\mathbb{C}^3$.

Extending this to $\mathbb{C}^2$ seems to be difficult as a linear projection would introduce nodal singularities in general.

There *are* examples of affine algebraic curves that do not embed *algebraically* into $\mathbb{A}^2$. One obstruction in this case is that for a smooth curve $X$ in $\mathbb{A}^2$, we have $\Omega^1_{X}$ is a trivial line bundle, whereas there *are* affine algebraic curves with non-trivial $\Omega^1_{X}$. However, this is *not* an obstruction in the case of an open Riemann surface since line bundles on it are *holomorphically* trivial!