He starts with x amount of money.

After the first doubling, he has:

2x - 16

After the second doubling, he has:

2(2x - 16) - 16 = 4x - 32 - 16 = 4x - 48

After the third doubling, he has:

2(4x - 48) - 16 = 8x - 96 - 16 = 8x - 112

After the forth doubling, he has:

2(8x - 112) - 16 = 16x - 224 - 16 = 16x - 240 ... at which point, he had 0 dollars.

Solving this sitution, we have:

16x - 240 = 0

16x = 240

x = 15