g(g(g(g(g(x))))) = 19.

Step 1:

Let g(g(g(g(x)))) = A, then g(A) = 19
If A is even, A = 19*2 = 38
If A is odd, A = 19-5 = 14, which is not odd. So A cannot be odd.
End of Step 2: g(g(g(g(x)))) = A = 38

Step 2:

A = g(g(g(g(x)))) = 38
Let g(g(g(x))) = B, then g(B) = 38
If B is even, B = 38*2 = 76
If B is odd, B = 38-5 = 33. So, B can be 76 or 33.
End of Step 2: g(g(g(x))) = B = 76 or 33

Step 3:

g(g(g(x))) = B = 76 or 33
Let g(g(x)) = C. We now have two cases
Case 1: g(C) = 76,
If C is even, C = 76*2 = 152
If C is odd, C = 76-5 = 71
Case 2:g(C) = 33,
If C is even, C = 33*2 = 66
If C is odd, C = 33-5 = 28, which is not odd. So A cannot be odd.
End of Step 3: C = g(g(x)) = 152 or 71 or 66

Step 4:

Let g(x) = D. We now have three cases

Case 1:

g(D) = 152
If D is even, D = 152*2 = 304
If D is odd, D = 152-5 = 147
Case 2: g(D) = 71
If D is even, D = 71*2 = 142
If D is odd, D = 71-5 = 66, which is not an odd number!
Case 3: g(D) = 66
If D is even, D = 66*2 = 132
If D is odd, D = 66-5 = 61
End of Step 4: g(x) = D = 61 or 132 or 142 or 147 or 304

Step 5:

Case 1: g(x) = 61
If x is even, x = 61*2,
If x is odd, x = 61-5 = 56, which is not an odd number!
Case 2: g(x) = 132
If x is even, x = 132*2,
If x is odd, x = 132-5 = 127
Case 3: g(x) = 142
If x is even, x = 142*2,
If x is odd, x = 142-5 = 137
Case 4: g(x) = 147
If x is even, x = 147*2,
If x is odd, x = 147-5 = 142, which is not an odd number!
Case 5: g(x) = 304
If x is even, x = 304*2,
If x is odd, x = 304-5 = 299

So, x = 299,304*2, 147*2, 137,132*2, 127, 122*2, 61*2 are the possible values for x which are 8 in number.

Answer 8

GMAT way:

General observation:
For every even value of g(Y),
If g(Y) = Even, Y takes two values, one Odd one Even
If g(Y) = odd, Y takes one value, Even.

Step 1:
g(g(g(g(g(x))))) = 19 = odd, so g(g(g(g(x)))) takes one value, EVEN.
Step 2:
g(g(g(g(x)))) = EVEN, So g(g(g(x))) takes two values one ODD(say O1), one EVEN(say E1).
Step 3:
If g(g(g(x))) = EVEN(E1), g(g(x)) takes two values one ODD(say O2), one EVEN(say E2).
If g(g(g(x))) = ODD(O1), g(g(x)) takes one value EVEN(say E3).
We have three values for g(g(x)): O2,E2,E3
Step 4:
If g(g(x)) = EVEN(E2), g(x) takes two values one ODD(say O3), one EVEN(say E4).
If g(g(x)) = EVEN(E3), g(x) takes two values one ODD(say O4), one EVEN(say E5).
If g(g(x)) = ODD(O1), g(x) takes one value EVEN(say E6).
We have three values for g(x): O3,O4,E4,E5,E6
Step 5:
If g(g(x)) = EVEN(E4), x takes two values one ODD(say O5), one EVEN(say E7).
If g(g(x)) = EVEN(E5), x takes two values one ODD(say O6), one EVEN(say E8).
If g(g(x)) = EVEN(E6), x takes two values one ODD(say O7), one EVEN(say E9).
If g(g(x)) = ODD(O3), x takes one value EVEN(say E10).
If g(g(x)) = ODD(O4), x takes one value EVEN(say E11).

So the possible values of x are O5,O6,O7,E7,E8,E9,E10,E11(8 in number)