What are examples of unexpected algebraic numbers of high degree occured in some math problems? Recently I asked a question about a possible transcendence of the number $\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)/\left(\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)\right)$, which, to my big surprise, turned out to be an algebraic number, but not some decent algebraic number like $\left(\sqrt{5}-1\right)/2$, but an enormous one with the minimal polynomial of degree 120 and a coefficient exceeding $10^{15}$.
So, my question: are there other interesting examples of numbers occurred in some math problems that were expected likely to be transcendental, but later unexpectedly were proven to be algebraic with a huge minimal polynomial.
 A: I don't know if Conway's constant is quite what you are looking for, as I'm not sure one would expect it initially to be transcendental or not. So, perhaps it's my bad intuition, but I was certainly surprised to learn that it is an algebraic number with minimal polynomial of degree 71. 
A: Some numbers were found recently algebraic using the LLL and PSLQ algorithms : these algorithms return directly the integer coefficients of a polynomial starting with the numerical value provided with enough precision. They are implemented in many Computer algebra software (for example lindep and algdep of pari/gp).
Broadhurst found that the third and fourth bifurcation points (B3 and B4) of the logistic map were algebraic of degree $12$ and $240$ (page 3 from this paper and 5 from this one and this ps file for B4).
See too this MO thread 'What Are Some Naturally-Occurring High-Degree Polynomials?'.
A: The areas of cyclic polygons is a great place to look for this sort of number.  Given the side lengths of a convex cyclic pentagon, you can find the area independent of how the sides are ordered.  But as Robbins found, you have to solve a seventh degree equation which in general cannot be expected to be reducible.  Succeed, and you still have to take the square root of your solution.  The seventh degree equation can be extended to cover cyclic hexagons, but beyond that you need degree 38 ... or more.
