In section 480 of his Treatise on Electricity and Magnetism Maxwell argues that the differential describing the work per pole strength around a current in an infinite wire is the differential of the function $2i\arctan (y/x) +C$. The differential of this function seems to be an exact differential, but the function is not entirely path independent, since the arctan function gives an infinite number of angle values.
My question is this: Is this an exception to the rule that exact differentials are path independent or am I mistakenly identifying this function as having an exact differential?
I think you're confusing exact with closed. The function $\arctan(y/x)$ is only smooth over simply connected subdomains of $\Bbb{R}^2\setminus\{(0,0)\}$. For instance, in an annulus around the origin it's not smoothly defined. So if you take the differential form $$\omega = -\frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2} dy$$ this differential form is exact on simply connected subdomains of $\Bbb{R}^2\setminus\{(0,0)\}$, being the differential of $\arctan(y/x)$, so $d\omega = 0$ on any such domain. This implies $d\omega = 0$ on the entire $\Bbb{R}^2\setminus\{(0,0)\}$. So it's closed. However, $d\omega$ is not exact on $\Bbb{R}^2\setminus\{(0,0)\}$ because there's no smooth definition for $\arctan(y/x)$ on this domain.