Let the initial amount of wine be $x$.
$x - 15 - 15 - 15 : 15+ 15 + 15 = 343: 169$
$x - 45 : 45 = 343:169$
$x = \frac{343 \times 45}{169} + 45$
$=45 \times \left[ \frac{343}{169} + 1\right]$
$=45 \times \frac{512}{169}$
$=136.33$
The above approach is wrong because after the first replacement, we are no longer having pure wine but a mixture of wine and water.
Amount of wine remaining $= \frac{343x}{343+169} = \frac{343x}{512}.$
Amount of wine after first add/remove $= x - 15 = x\left(1-\frac{15}{x}\right)$, after second add/remove $=x\left(1-\frac{15}{x}\right)^2$ and similarly after the third add/remove $x\left(1-\frac{15}{x}\right)^3$.
Thus we get $\frac{343x}{512} = x\left(1-\frac{15}{x}\right)^3$
$\implies \frac{7}{8} = 1 - \frac{15}{x}$
$\implies \frac{15}{x} = \frac{1}{8}$
$\implies x = 15 \times 8 = 120.$