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.
Best Answer
The conjugacy problem in $SL(2,Z)$. For matrices $M \in GL(2,Z)$ having trace of absolute value $>2$, the slope of its expanding eigenvector has an eventually periodic continued fraction expansion (it is a quadratic irrational), and the primitive period loop is a conjugacy invariant in $SL(2,Z)$. Throw in the absolute value of the trace itself and you have a complete conjugacy invariant in $SL(2,Z)$.