Continued fractions are a well-known method for producing rational approximations to given real numbers, which can be shown to be optimal with an appropriate optimality measure.
Does there exist a similar method for producing a best algebraic approximation to a given real (or complex) number? As with rational numbers, the algebraic numbers are dense in the complexes, so it is important to bound the "complexity" of the approximation if one hopes to get a definite answer, but since the approximation method will dictate this choice to some degree, I will be vague about the actual complexity measure.
You might be interested in integer relation algorithms. These take high-precision approximations to some real numbers of interest and try to find small integer coefficients (with a suitable definition of small) such that the given numbers combine to something that might be zero within the uncertainty caused by the limited working precision.
To find a polynomial with integer coefficients that has among its roots a given, supposedly algebraic, number $x$ of degree less than or equal to $n$, one would feed high-precision approximations of $(1, x, x^2, \ldots, x^n)$ to such an algorithm.
Pari/GP uses an LLL-based integer relation algorithm as part of its
lindepandalgdepfunctions.