Certainly there is one type of study of this question, coming under the heading of the Markoff Spectrum, going back to the German tranliteration of Markov. There is a very nice book called "The Markoff and Lagrange Spectra" by Thomas W. Cusick and Mary E. Flahive. It is this topic that lead to the Markoff Numbers, which are any of the triple $x,y,z$ of a positive integer solution to $$ x^2 + y^2 + z^2 = 3 x y z . $$ 

If you are curious, there is my paper with Kaplansky in the Illinois Journal, pdf at:
http://zakuski.math.utsa.edu/~jagy/bib.html 

I am especially proud of section 8.3, called "Other Families," as I was able to construct a mirror image of our "Markov Ratios" 
$$ 9 - \frac{4}{m^2} $$
in forms that gave
$$ 9 + \frac{4}{m^2} $$

To clarify, any indefinite binary quadratic form with integer coefficients and non-square discriminant has a nonzero "minimum" which is the smallest nonzero absolute value of any value obtained with integral values for the arguments. Our "Markov Ratio" was simply the discriminant divided by the square of this minimum. 

Well, your Golden Ratio has the samllest possible Markov Ratio, with 5, the form being equivalent to $x^2 + x y - y^2.$ The second smallest is 8, form $2 x^2 + 4 x y - 2 y^2$ which is visibly imprimitive.

If you look at the Cusick and Flahive book, you will see how the Markoff and Lagrange spectra coincide with "Markov Ratio" below 9 but are rather different above, where the description using binary forms is no longer adequate.

For the hasty:
http://en.wikipedia.org/wiki/Markov_spectrum