Equation with several exponents

Try

NSolve[x^x^(x^(1/6)/6) == Sqrt[2^9]^(2*Sqrt[2])^(2/3 + 2*Sqrt[2]), x, Reals]
(*{x->512}*)

This is essentially the same as Ulrich's solution but provides the motivation for restricting the domain of Solve

eqn = x^x^(x^(1/6)/6) == Sqrt[2^9]^(2*Sqrt[2])^(2/3 + 2*Sqrt[2]) // 
  Simplify

(* x^x^(x^(1/6)/6) == 8^(3 8^Sqrt[2]) *)

The RHS of eqn is real

Element[eqn[[-1]], Reals]

(* True *)

Consequently, finding the FunctionDomain for the LHS of eqn

const = FunctionDomain[eqn[[1]], x] // Simplify

(* x > 0 *)

Then,

Solve[eqn && const, x][[1]]

(* {x -> 512} *)

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