Find the value of the following expression:

$$\left| \, |{ -|{-1 + 1}| - 1 }| + 1\right|.$$
To evaluate this expression, I need to follow the order of operations and simplify the innermost absolute value bars first. I can rewrite the expression as follows: $$\left| \, |{ -|{-1 + 1}| - 1 }| + 1\right| = \left| \, |{ -|{0}| - 1 }| + 1\right|.$$ Since the absolute value of zero is zero, I can simplify further: $$\left| \, |{ -|{0}| - 1 }| + 1\right| = \left| \, |{ -0 - 1 }| + 1\right| = \left| \, |{ -1 }| + 1\right|.$$ The absolute value of -1 is 1, so I can simplify even more: $$\left| \, |{ -1 }| + 1\right| = \left| \, 1 + 1\right| = \left| \, 2\right|.$$ 