The largest real value of $$ a $$ for which the equation $$ |x + a| + |x - 1| = 2 $$ has an infinite number of solutions for $$ x $$ is:

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To solve this problem:

1. The equation $$ |x + a| + |x - 1| = 2 $$ implies that the sum of distances between the points $$ x $$, $$ -a $$, and $$ 1 $$ must be equal to 2.

2. For infinite solutions to exist, the distance between $$ -a $$ and $$ 1 $$ must be exactly equal to 2 because any deviation would limit the number of solutions.

3. Consider the following cases:
   - If $$ x $$ lies between $$ -a $$ and $$ 1 $$, the sum of distances is exactly 2.
   - For infinite solutions, $$ -a $$ and $$ 1 $$ must be separated by exactly 2 units.

4. Solve for $$ a $$:  
   - Distance between $$ -a $$ and $$ 1 = 2 $$:  
     $$ |-a - 1| = 2 $$  
     Solving gives:  
     $$ a = 1 $$

Thus, the maximum value of $$ a $$ for infinite solutions is **1**.

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