In a convex heptagon, the degree measures of the interior angles are $x, ~x, ~x-2, ~x-2,
~x + 2, ~x + 2$ and $x + 4$ degrees. What is the degree measure of the largest interior angle?
The sum of the angle measures in a polygon with 7 angles is $180(7-2) = 900$ degrees. Therefore, we must have \[x + x + (x-2) + (x-2) + (x+2) + (x+2) + (x+4) = 900.\] Simplifying the left side gives $7x + 4 = 900$, so $7x = 896$ and $x = 128$.  Therefore, the measure of the largest interior angle is $x + 4 = \boxed{132}$ degrees.