c = 6; (*constant adder in square root*)
b = 6; (*initial value (for F_0)*)
f0[x_] := x;

(*finding f(x)*)
N1 = 10; (*precision*)
g[x_] := Sqrt[c + x];
f[x_] := Nest[g, f0[x^2^N1], N1];

(*findinf X*)
N2 = 10; (*precision*)
res1 = FindRoot[f[x] == b, {x, c}, WorkingPrecision -> 1000];
X = x /. res1;

F[n_] := f[X^2^n]

(* you can use this function for comparing (this for eq1) *)
F2[n_] := F2[n - 1]^2 - c;
F2[0] := b;

F[0]
F[-1]
F[-2]

6.00000000000000000000000000000000000000000000000000000000000000000000...
3.46410161513775458705489268301174473388561050762076125611161395890386...
3.07637800264170309696602586393672241931859085772505962544063421316756...