What are examples of unexpected algebraic numbers of high degree occured in some math problems?

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.

Sometimes you can get unexpected algebraic values while working with hypergeometric functions. For example, the following absolute value of a complex-valued $_4F_3$ function: $$\left|\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\sqrt{\phi }\right)\right|,$$ where $\phi$ is the golden ratio, is actually an algebraic number with the minimal polynomial of degree 80 and a coefficient exceeding $10^{55}$: $$340282366920938463463374607431768211456 x^{80}+152118072027387528179604384645120000000000 x^{72}+202824096036516704239472512860160000000000 x^{70}-45334718235548594051481600000000000000000000 x^{62}+15111572745182864683827200000000000000000000 x^{60}-5629499534213120000000000000000000000000000000 x^{56}-24769797950537728000000000000000000000000000000 x^{54}-33776997205278720000000000000000000000000000000 x^{52}-9007199254740992000000000000000000000000000000 x^{50}+1006632960000000000000000000000000000000000000000 x^{46}+3523215360000000000000000000000000000000000000000 x^{44}+74161139200000000000000000000000000000000000000000 x^{40}+300000000000000000000000000000000000000000000000000 x^{38}+675000000000000000000000000000000000000000000000000 x^{36}+1050000000000000000000000000000000000000000000000000 x^{34}-2975290298461914062500000000000000000000000000000000 x^{32}-14701161193847656250000000000000000000000000000000000 x^{30}-37252902984619140625000000000000000000000000000000000 x^{28}-74505805969238281250000000000000000000000000000000000 x^{26}-59604644775390625000000000000000000000000000000000000 x^{24}-22351741790771484375000000000000000000000000000000000 x^{22}+7450580596923828125000000000000000000000000000000000 x^{20}-555111512312578270211815834045410156250000000000000000 x^{16}-1110223024625156540423631668090820312500000000000000000 x^{14}-1665334536937734810635447502136230468750000000000000000 x^{12}-2220446049250313080847263336181640625000000000000000000 x^{10}+82718061255302767487140869206996285356581211090087890625$$

I'm not sure if it is expressible in radicals.