A badly approximable irrational is one whose continued fraction denominators are bounded; equivalently, if $\alpha$ is badly approximable then there is a $c(\alpha) > 0$ such that $$c(\alpha) = \liminf_{q \to \infty}q|q\alpha-p|$$ Where $p$ is the nearest integer to $q\alpha$. The constant $c(\alpha)$ is said to be the approximation constant to $\alpha$.
My question: what is known about $c(x)$? Do we know its value for particular $x$? The Wolfram Mathworld page has no information on it.
The approximation constant $ c ( x ) $ is also known as the Lagrange constant (see "Markoff–Lagrange spectrum and extremal numbers" by D. Roy).
Sometimes it is defined as its inverse instead (see "The role of the Lagrange constant in some nonlinear waves equations" by A. K. Ben-Naoum).
An example, which now appears in the Wolfram Mathworld page you mentioned, is $ c ( 1 / \sqrt{5} ) = ( \sqrt{5} - 1 ) / 2 $.