If $\alpha$ is a real irrational number, then there are infinitely many coprime integers $p,q$ with $q > 0$ such that
$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^2}$$
by Dirichlet's theorem.
One can easily construct a real number such that one can replace the right hand side with a function of $q$ that goes to zero faster than $q^{-2}$ by an arbitrary amount. That is, for any function $f$ with $\lim_{x \rightarrow \infty} f(x) = 0$ we can find a real number $\alpha_f$ such that there exist infinitely many integers $p,q$ with $\gcd(p,q) = 1, q > 0$ satisfying
$$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{f(q)}{q^2}.$$
One can do this in at least two ways. First, one can simply choose the sequence of partial quotients in the continued fraction expansion of $\alpha$ to tend to infinity arbitrarily fast. Essentially equivalently, one can choose
$$\displaystyle \alpha = \sum_{n=1}^\infty a_n$$
with the sequence $\{a_n\}_{n \geq 1} \subset \mathbb{R}_{>0}$ tending to zero arbitrarily fast. For example, Liouville's original construction of a transcendental number had the choice $a_n = 10^{-n!}$.
Conversely, the badly approximable numbers are those whose sequence of partial quotients is bounded. The absolute worst approximable number is therefore the unique real number whose sequence of partial quotients consist of only 1's. This is the Golden ratio, and the statement that this is the worst approximable irrational number is due to Hurwitz. In particular, Hurwitz proved that one can improve the constant 1 in the numerator of Dirichlet's theorem by any number greater than $1/\sqrt{5}$, but no smaller.