1. **State the problem:** We are given a custom operation defined as $a \triangledown b = 2a - b$. We need to find the value of $\frac{1}{4} \triangledown \left( \frac{3}{4} \triangledown 1 \right)$. 2. **Understand the operation:** The operation $a \triangledown b$ means we take twice $a$ and subtract $b$. So for any numbers $a$ and $b$, $a \triangledown b = 2a - b$. 3. **Calculate the inner operation:** First, calculate $\frac{3}{4} \triangledown 1$. Using the formula, this is: $$\frac{3}{4} \triangledown 1 = 2 \times \frac{3}{4} - 1 = \frac{6}{4} - 1 = \frac{6}{4} - \frac{4}{4} = \frac{2}{4} = \frac{1}{2}.$$ 4. **Calculate the outer operation:** Now calculate $\frac{1}{4} \triangledown \left( \frac{1}{2} \right)$: $$\frac{1}{4} \triangledown \frac{1}{2} = 2 \times \frac{1}{4} - \frac{1}{2} = \frac{2}{4} - \frac{1}{2} = \frac{1}{2} - \frac{1}{2} = 0.$$ 5. **Final answer:** The value of $\frac{1}{4} \triangledown \left( \frac{3}{4} \triangledown 1 \right)$ is $0$.