Some answers from Applications of finite continued fractions in fact are Applications of periodic continued fractions. I think that it should be separate question.
What can you add to the following list of applications?
1) Calculation and approximation of quadratic irrational numbers. calculation of corresponding covex hulls.
2) Pell equation and calculation of fundamental units in quadratic fields.
3) Reduction of quadratic forms. Calculation of class numbers of imaginary quadratic field.
4) Legendre's factorization method.