It is given that $|z - 1 + i| = 2$ - HSC - SSCE Mathematics Extension 2 - Question 7 - 2024 - Paper 1

The center of the circle given by the expression is at the point (1, -1) in the complex plane, and it has a radius of 2.

Next, we calculate the distance from the origin (0, 0) to the center of the circle (1, -1). The distance is given by the formula:
$d = ext{sqrt}((1 - 0)^2 + (-1 - 0)^2) = \sqrt{1^2 + (-1)^2} = \sqrt{2}.$

To find the maximum distance from the origin to the circle's perimeter, we add the radius to the distance to the center:
$\text{Maximum } |z| = d + r = \sqrt{2} + 2.$

Thus, we can express the maximum possible value of $|z|$ as:
$|z| = 2 + \sqrt{2}.$

From the provided options, the maximum possible value of $|z|$ is therefore option C: $2 + \sqrt{2}$.