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We construct continued fraction expansions for several families of theLaurent series in $\mathbb{Q}[[t^{-1}]]$. To the best of the author'sknowledge, this is the first result of this kind since Gauss derived thecontinued fraction expansion for $(1+t)^r$, $r\in\mathbb{Q}$ in 1813. As anapplication, we apply an analogue of the hypergeometric method to one of thosefamilies and derive non-trivial lower bounds on the distance $|x -\frac{p}{q}|$ between one of the real roots of $3x^3 - 3tx^2-3ax+at$,$a,t\in\mathbb{Z}$ and any rational number, under relatively mild conditions onthe parameters $a$ and $t$. We also show that every cubic irrational$x\in\mathbb{R}$ admits a (generalised) continued fraction expansion in aclosed form that can be explicitly computed. Finally, we provide an infiniteseries of cubic irrationals $x$ that have arbitrarily (but finitely) manybetter-than-expected rational approximations. That is, they are such that forany $\tau< 3+\frac{15\ln 2}{24}\approx 3.4332...$ the inequality $||qx|| <(H(x)^{\tau} qe^{c\sqrt{\ln q}})^{-1}$ has many solutions in integer $q$.

Continued fractions of cubic Laurent series