Sometimes, when getting some numerical results when investigating number theory sequences with a computer, I find myself suspecting that a decimal value ($a$) I have found might be a quadratic irrational. My usual next step is to look at the continued fraction for $a$ and hopefully find that it is periodic, and then I can find a closed expression for $a$.
My question stems from my inability (so far) to do anything along similar lines when I find (by calculating examples) a value $b$ that I suspect might be a cubic irrational such as $\frac{2^{1/3} + 5.2^{2/3}}{7}$. I believe that it is unknown whether there is any pattern for such irrationals when expressed as continued fractions (and maybe not even known if the terms are bounded?).
Does anyone have any ideas about how to investigate such a decimal with a view to identifying exactly which cubic irrational expression might be the an appropriate closed form? Are there any other techniques better than continued fractions?