When people write $\mathbb C((t))=\mathbb C[[t]][t^{-1}]$, what exactly do they mean?
Do they mean that the field of formal Laurent series $\mathbb C((t))$ is a polynomial ring in the variable $t^{-1}$ with values in the ring $\mathbb C[[t]]$ of formal power series or do they simply denote the fact that $\mathbb C((t))$ is localized at the multiplicative set generated by $t$? Every element in $\mathbb C((t))$ can be represented as a polynomial in $t^{-1}$ with coefficients in $\mathbb C[[t]]$, so it seems to make sense that we view $\mathbb C((t))$ as a polynomial ring with values in $\mathbb C[[t]]$. Does anyone disagree with such a statement?
It means the localization. The polynomial ring would be strictly larger, and in particular would not impose the relation $t \cdot t^{-1} = 1$.