One way is:
n = 20000;
th = \[Pi]/7;
$MaxExtraPrecision = 10000;(* Increase this value if necessary.*)
N[LegendreP[n, th], 20]
(*-0.0059312818372221032034*)
Second way:
n = 2000000;
th = \[Pi]/7;
LegendrePPP[n_, z_] := (1/Pi) NIntegrate[(z - Sqrt[z^2 - 1] Cos[t])^n, {t, 0, Pi},
WorkingPrecision -> 30, MaxRecursion -> 20]
LegendrePPP[n, th]
(*0.000475301944163975616743822092080*)
Third way is the best it's very fast:
ClearAll["`*"]; Remove["`*"];
S[n_, z_] = (Series[LegendreP[n, z], {n, Infinity, 1}] // Normal // Simplify)
n = 10000000;(* Try: n = 10^100 *)
th = \[Pi]/7;
N[S[n, th], 20]
(*-0.00018112363719941377947*)