Some high-degree polynomials have been discovered using integer-relation algorithms applied to the powers of interesting constants. Here is a quotation from Section 3 of a paper of David Bailey, which can be found here: http://www.nersc.gov/homes/dhbailey/dhbpapers/pslq-cse.pdf

"One of the first results of this sort was the identification of the constant B3 = 3.54409035955 · · · [1]. B3 is the third bifurcation point of the logistic map $x_{k+1} = rx_k(1 − x_k )$, which exhibits period doubling shortly before the onset of chaos. To be precise, B3 is the smallest value of the parameter r such that successive iterates $x_k$ exhibit eight-way periodicity instead of four-way periodicity. Computations using a predecessor algorithm to PSLQ found that B3 is a root of the polynomial

$0 = 4913 + 2108t^2 − 604t^3 − 977t^4 + 8t^5 + 44t^6 + 392t^7 − 193t^8 − 40t^9 + 48t^{10} − 12t^{11} + t^{12}.$

Recently, B4 = 3.564407268705 · · ·, the fourth bifurcation point of the logistic map, was identified using PSLQ by British physicist David Broadhurst [5]. Some conjectural reasoning had suggested that B4 might satisfy a 240-degree polynomial, and some further analysis had suggested that the constant α = −B4 (B4 − 2) might satisfy a 120-degree polynomial. In order to test this hypothesis, Broadhurst applied a PSLQ program to the 121-long vector $(1, α, α^2 , · · · , α^{120})$. Indeed, a relation was found, although 10,000 digit arithmetic was required. The recovered integer coefficients descend monotonically from $257^{30} ≈ 1.986 × 10^{72}$ to one."

Some high-degree polynomials have been discovered using integer-relation algorithms applied to the powers of interesting constants. Here is a quotation from Section 3 of a paper of David Bailey, which can be found here: http://www.nersc.gov/homes/dhbailey/dhbpapers/pslq-cse.pdf

"One of the first results of this sort was the identification of the constant B3 = 3.54409035955 · · · [1]. B3 is the third bifurcation point of the logistic map $x_{k+1} = rx_k(1 − x_k )$, which exhibits period doubling shortly before the onset of chaos. To be precise, B3 is the smallest value of the parameter r such that successive iterates $x_k$ exhibit eight-way periodicity instead of four-way periodicity. Computations using a predecessor algorithm to PSLQ found that B3 is a root of the polynomial

$$0 = 4913 + 2108t^2 − 604t^3 − 977t^4 + 8t^5 + 44t^6 + 392t^7 − 193t^8 − 40t^9 + 48t^{10} − 12t^{11} + t^{12}$$

Or, if $x=-t(t - 2)$,

$$0 = 4913 + 527 x^2 - 188 x^3 + 47 x^4 - 12 x^5 + x^6$$

which is a sextic with a non-solvable Galois group. Recently, B4 = 3.564407268705 · · ·, the fourth bifurcation point of the logistic map, was identified using PSLQ by British physicist David Broadhurst [5]. Some conjectural reasoning had suggested that B4 might satisfy a 240-degree polynomial, and some further analysis had suggested that the constant $α = −B4 (B4 − 2)$ might satisfy a 120-degree polynomial. 

In order to test this hypothesis, Broadhurst applied a PSLQ program to the 121-long vector $(1, α, α^2 , · · · , α^{120})$. Indeed, a relation was found, although 10,000 digit arithmetic was required. The recovered integer coefficients descend monotonically from $257^{30} ≈ 1.986 × 10^{72}$ to one."