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 $a_n = 10^{-n!}$.