Every real number can be approximated by an ever-closer series of rationals, since the rationals are dense in the real numbers (e.g. the square root of two is approximated by the fractions 1/1, 14/10, 141/100, 1414/1000, 14142/10000, ...). The question is how good of an approximation you can get for rational numbers with denominators up to some bound.
This turns out to be related to the size of the terms in the continued fraction expansion of a number - larger terms in the continued fraction expansion yield better approximations of your chosen value. (The unusually-good-for-its-complexity rational approximation of pi as 355/113 exists because of an unusually large term early in the continued fraction expansion of pi as [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, ...]. If we round the 1/292 term portion of the continued fraction to zero, we get 3+1/(7+1/(15+1/1)) = 355/113.)
The golden ratio's continued fraction expansion is all 1's, so in this sense it has the worst rational approximations of any real number. In practice, people tend to care about the asymptotics of this approximation quality, which leads to the notion of https://mathworld.wolfram.com/IrrationalityMeasure.html. More concrete implications between the terms of the continued fraction and the badness of rational approximations in this sense are mentioned at this MathOverflow thread: https://mathoverflow.net/questions/89600/numbers-with-known-irrationality-measures
Every real number can be approximated by an ever-closer series of rationals, since the rationals are dense in the real numbers (e.g. the square root of two is approximated by the fractions 1/1, 14/10, 141/100, 1414/1000, 14142/10000, ...). The question is how good of an approximation you can get for rational numbers with denominators up to some bound.
This turns out to be related to the size of the terms in the continued fraction expansion of a number - larger terms in the continued fraction expansion yield better approximations of your chosen value. (The unusually-good-for-its-complexity rational approximation of pi as 355/113 exists because of an unusually large term early in the continued fraction expansion of pi as [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, ...]. If we round the 1/292 term portion of the continued fraction to zero, we get 3+1/(7+1/(15+1/1)) = 355/113.)
The golden ratio's continued fraction expansion is all 1's, so in this sense it has the worst rational approximations of any real number. In practice, people tend to care about the asymptotics of this approximation quality, which leads to the notion of https://mathworld.wolfram.com/IrrationalityMeasure.html. More concrete implications between the terms of the continued fraction and the badness of rational approximations in this sense are mentioned at this MathOverflow thread: https://mathoverflow.net/questions/89600/numbers-with-known-irrationality-measures