As it stands it will not work because it has circular dependencies (setting a === A, otherwise a remains undefined)

In[134]:= Clear[x, a, z]
z[n_Integer] := n Log[x[n]] + x[n]^(1/3)
a[n_Integer] := Cos[z[n]]
x[n_Integer] := 120 - Exp[n] - a[n] - 1

In[141]:= z[3]
$RecursionLimit::reclim: Recursion depth of 1024 exceeded. >>

If it's decoupled by free will (taking a as different from A and giving a a random definition) and the values of A are searched for it is simply

In[142]:= Clear[x, a, z, A]
z[n_Integer] := n Log[x[n]] + x[n]^(1/3)
A[n_Integer] := Cos[z[n]]
x[n_Integer] := 120 - Exp[n] - a[n] - 1
a[n_Integer] := n  (* ? *)

In[152]:= A /@ Range[7] // N
Out[152]= {-0.982096, -0.042459, 0.835536, 0.0925525, 
 4.94991*10^7 - 2.50804*10^7 I, 2.22449*10^10 + 8.36742*10^9 I, -9.6318*10^12 - 1.82179*10^12 I}

no need for a loop. Of course you get out other values too in the same way

In[153]:= a /@ Range[7]
Out[153]= {1, 2, 3, 4, 5, 6, 7}

In[154]:= x /@ Range[7] // N
Out[154]= {115.282, 109.611, 95.9145, 60.4018, -34.4132, -290.429, -984.633}