Let $A(0, 1)$, $B(0, 0)$, $C(1, 0)$, and $D(1, 1)$ be the coordinates of the vertices of square $ABCD$.

The point $X$ on side $BC$ is given by $BX = \frac{3}{4}$, thus the coordinates of point $X$ are $(0.75, 0)$.

The laser beam starts at $A(0, 1)$ and hits $X(0.75, 0)$. We can calculate the slope of the line from $A$ to $X$:

$\text{slope} = \frac{0 - 1}{0.75 - 0} = -\frac{4}{3}$

The equation of the line from $A$ to $X$ is:

$y - 1 = -\frac{4}{3}(x - 0)$ $y = -\frac{4}{3}x + 1$

To find where this line intersects the sides of the square, we will evaluate it at the sides:

1. Intersection with $BC$ (where $x = 0$): $y = -\frac{4}{3}(0) + 1 = 1$ This point $(0, 1)$ is actually $A$, which we already know.

2. Intersection with $CD$ (where $y = 0$): $0 = -\frac{4}{3}x + 1$ Solving for $x$: $\frac{4}{3}x = 1 \implies x = \frac{3}{4}$ The intersection point is $\left(\frac{3}{4}, 0\right)$.

3. Now, we reflect the square across the sides. To visualize the path of the laser, we will consider "unfolding" the square. Reflect the square across the line $BC$ to create a new square where the line continues in a straight path.

   The new vertex is at $C'(1, 0)$, and since the laser hits $X(0.75, 0)$ and travels down to $D'$ (reflection across $x = 1$):

   • Reflect across line $y = 0$ to find $D' = (1, -1)$.

The laser now travels from $A(0, 1)$ to $X(0.75, 0)$, then it continues from $X$ to $D'$.

Now, we find the total distance:

1. Distance from $A(0, 1)$ to $X(0.75, 0)$: $d_1 = \sqrt{(0.75 - 0)^2 + (0 - 1)^2} = \sqrt{(0.75)^2 + (-1)^2} = \sqrt{0.5625 + 1} = \sqrt{1.5625} = \frac{5}{4}$

2. Distance from $X(0.75, 0)$ to $D'(1, -1)$: $d_2 = \sqrt{(1 - 0.75)^2 + (-1 - 0)^2} = \sqrt{(0.25)^2 + (-1)^2} = \sqrt{0.0625 + 1} = \sqrt{1.0625} = \frac{15}{16}$

Now, we calculate the total path length: $d_{\text{total}} = d_1 + d_2 = \frac{5}{4} + \frac{15}{16}$

To add these fractions, convert $\frac{5}{4}$ into sixteenths: $\frac{5}{4} = \frac{20}{16}$ Thus, $d_{\text{total}} = \frac{20}{16} + \frac{15}{16} = \frac{35}{16}$

Finally, the length of the total path of the laser beam is

$\boxed{\frac{35}{16}}$