A  triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle?
1. **Identify the possible integer side lengths:**
   Given the perimeter of the triangle is 7, we need to find sets of three positive integers \(a, b, c\) such that \(a + b + c = 7\) and they satisfy the triangle inequality \(a + b > c\), \(a + c > b\), and \(b + c > a\).

2. **List all combinations of integers that sum to 7:**
   - \(1 + 1 + 5\) (not a valid triangle since \(1 + 1 \not> 5\))
   - \(1 + 2 + 4\) (not a valid triangle since \(1 + 2 \not> 4\))
   - \(1 + 3 + 3\) (valid triangle)
   - \(2 + 2 + 3\) (valid triangle)

3. **Calculate the area of each valid triangle using Heron's formula:**
   Heron's formula states that the area \(A\) of a triangle with sides \(a, b, c\) is given by:
   \[
   A = \sqrt{s(s-a)(s-b)(s-c)}
   \]
   where \(s\) is the semi-perimeter of the triangle, \(s = \frac{a + b + c}{2}\).

4. **Calculate the area for the triangle with sides \(1, 3, 3\):**
   - Semi-perimeter \(s = \frac{1 + 3 + 3}{2} = 3.5\)
   - Area \(A = \sqrt{3.5(3.5-1)(3.5-3)(3.5-3)} = \sqrt{3.5 \times 2.5 \times 0.5 \times 0.5}\)
   \[
   A = \sqrt{3.5 \times 2.5 \times 0.25} = \sqrt{2.1875} \approx 1.48
   \]

5. **Calculate the area for the triangle with sides \(2, 2, 3\):**
   - Semi-perimeter \(s = \frac{2 + 2 + 3}{2} = 3.5\)
   - Area \(A = \sqrt{3.5(3.5-2)(3.5-2)(3.5-3)} = \sqrt{3.5 \times 1.5 \times 1.5 \times 0.5}\)
   \[
   A = \sqrt{3.5 \times 1.5 \times 1.5 \times 0.5} = \sqrt{3.9375} \approx 1.98
   \]

6. **Compare the areas:**
   - The area of the triangle with sides \(1, 3, 3\) is approximately \(1.48\).
   - The area of the triangle with sides \(2, 2, 3\) is approximately \(1.98\).

Since \(1.98 > 1.48\), the triangle with sides \(2, 2, 3\) has the maximum possible area.

The final answer is \( \boxed{ \frac{3\sqrt{7}}{4} } \)