The golden ratio $\phi=\frac{1+\sqrt5}2$ is sometimes said to be one of the most difficult numbers to approximate with rational numbers, because its continued fraction development $$\phi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cdots}}}$$has all entries equal to $1$, so that its convergents are as far from $\phi$ as possible. This can be quantified and made precise (I guess: I'd love to know the precisely how!) It shares its properties with all the numbers $\frac{a+b\phi}{c+d\phi}$ with $\left(\begin{smallmatrix}a&b\\\c&d\end{smallmatrix}\right)\in\mathrm{SL}(2,\mathbb Z)$ (and no others?)

Are there other numbers which are characterized by such extremal properties? Is there a second worst aproximable number?

The golden ratio $\phi=\frac{1+\sqrt5}2$ is sometimes said to be one of the most difficult numbers to approximate with rational numbers, because its continued fraction development $$\phi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cdots}}}$$has all entries equal to $1$, so that its convergents are as far from $\phi$ as possible. This can be quantified and made precise (I guess: I'd love to know the precisely how!) It shares this property with all the numbers $\frac{a+b\phi}{c+d\phi}$ with $\left(\begin{smallmatrix}a&b\\\c&d\end{smallmatrix}\right)\in\mathrm{SL}(2,\mathbb Z)$ (and no others?)

Are there other numbers which are characterized by such extremal properties? Is there a second worst aproximable number?