Diophantine approximation asks for the best approximation of a real number by rationals (relative to the size of the denominator). The basic paradigm has many similarities with Kolmogorov complexity, in fact, it can be seen as a version of complexity with very limited computational power. I will illustrate these similarities with the help of two classic results in Diophantine approximation: the Jarnik-Besicovitch theorem in metric Diophantine approximation and Thue's theorem on approximation of algebraic numbers.