Compute the smallest positive integer $a$ for which $$\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7$$
To solve the problem, we need to find the smallest positive integer \( a \) such that:
\[ \sqrt{a + \sqrt{a + \sqrt{a + \ldots}}} - \frac{1}{a + \frac{1}{a + \frac{1}{a + \ldots}}} > 7 \]

1. **Simplify the Nested Radicals and Fractions:**
   Let's denote:
   \[ x = \sqrt{a + \sqrt{a + \sqrt{a + \ldots}}} \]
   By the definition of the nested radical, we have:
   \[ x = \sqrt{a + x} \]
   Squaring both sides, we get:
   \[ x^2 = a + x \]
   Rearranging, we obtain:
   \[ x^2 - x - a = 0 \]
   Solving this quadratic equation for \( x \), we use the quadratic formula:
   \[ x = \frac{1 \pm \sqrt{1 + 4a}}{2} \]
   Since \( x \) must be positive, we take the positive root:
   \[ x = \frac{1 + \sqrt{1 + 4a}}{2} \]

2. **Simplify the Nested Fractions:**
   Let's denote:
   \[ y = a + \frac{1}{a + \frac{1}{a + \ldots}} \]
   By the definition of the nested fraction, we have:
   \[ y = a + \frac{1}{y} \]
   Multiplying both sides by \( y \), we get:
   \[ y^2 = ay + 1 \]
   Rearranging, we obtain:
   \[ y^2 - ay - 1 = 0 \]
   Solving this quadratic equation for \( y \), we use the quadratic formula:
   \[ y = \frac{a \pm \sqrt{a^2 + 4}}{2} \]
   Since \( y \) must be positive, we take the positive root:
   \[ y = \frac{a + \sqrt{a^2 + 4}}{2} \]

3. **Set Up the Inequality:**
   We need to satisfy the inequality:
   \[ \frac{1 + \sqrt{1 + 4a}}{2} - \frac{2}{a + \sqrt{a^2 + 4}} > 7 \]

4. **Simplify the Inequality:**
   Let's simplify the left-hand side:
   \[ \frac{1 + \sqrt{1 + 4a}}{2} - \frac{2}{a + \sqrt{a^2 + 4}} > 7 \]
   Multiply through by 2 to clear the denominators:
   \[ 1 + \sqrt{1 + 4a} - \frac{4}{a + \sqrt{a^2 + 4}} > 14 \]
   Rearrange to isolate the square root term:
   \[ \sqrt{1 + 4a} - \frac{4}{a + \sqrt{a^2 + 4}} > 13 \]

5. **Estimate \( a \):**
   To find the smallest \( a \) that satisfies this inequality, we can test values of \( a \) starting from a reasonable guess. Let's test \( a = 43 \):
   \[ \sqrt{1 + 4 \cdot 43} = \sqrt{173} \approx 13.15 \]
   \[ a + \sqrt{a^2 + 4} = 43 + \sqrt{43^2 + 4} = 43 + \sqrt{1849 + 4} = 43 + \sqrt{1853} \approx 43 + 43.05 = 86.05 \]
   \[ \frac{4}{86.05} \approx 0.046 \]
   \[ 13.15 - 0.046 \approx 13.104 \]
   Since \( 13.104 > 13 \), \( a = 43 \) satisfies the inequality.

The final answer is \( \boxed{43} \).