Consider the set $A = \{1, 2, 3, ..., 2008\}$. We say that a set is of [i]type[/i] $r, r \in \{0, 1, 2\}$, if that set is a nonempty subset of $A$ and the sum of its elements gives the remainder $r$ when divided by $3$. Denote by $X_r, r \in \{0, 1, 2\}$ the class of sets of type $r$. Determine which of the classes $X_r, r \in \{0, 1, 2\}$,  is the largest.
1. **Define the set and the problem:**
   Consider the set \( A = \{1, 2, 3, \ldots, 2008\} \). We need to determine which of the classes \( X_r \) (where \( r \in \{0, 1, 2\} \)) is the largest. A set is of type \( r \) if the sum of its elements gives the remainder \( r \) when divided by 3.

2. **Symmetry and Initial Observations:**
   By symmetry, we can initially assume that the sizes of \( X_1 \) and \( X_2 \) are equal. We will use induction to show that \( X_1 \) is the largest for the set \( A \).

3. **Base Case:**
   Consider the set \( B = \{1, 2, 3\} \):
   - The subsets of \( B \) are: \(\{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\).
   - The sums of these subsets modulo 3 are: \(1, 2, 0, 0, 1, 2, 0\).
   - Therefore, \( |X_0| = 3 \), \( |X_1| = 2 \), and \( |X_2| = 2 \).

4. **Inductive Step:**
   Assume for a set \( B = \{1, 2, \ldots, 3k\} \), the sizes of \( X_0 \), \( X_1 \), and \( X_2 \) are such that \( |X_0| \) is the largest. Now consider adding the next three elements \( 3k+1, 3k+2, 3k+3 \) to form the set \( B' = \{1, 2, \ldots, 3k+3\} \).

5. **Adding New Elements:**
   - Adding \( 3k+1 \) shifts the sums of subsets of \( B \) by 1 modulo 3.
   - Adding \( 3k+2 \) shifts the sums of subsets of \( B \) by 2 modulo 3.
   - Adding \( 3k+3 \) does not change the sums modulo 3.

6. **Counting New Subsets:**
   - Each new element \( 3k+1, 3k+2, 3k+3 \) will create new subsets that shift the sums of the original subsets.
   - The new subsets will be counted as follows:
     - Subsets including \( 3k+1 \) will contribute to \( X_1 \).
     - Subsets including \( 3k+2 \) will contribute to \( X_2 \).
     - Subsets including \( 3k+3 \) will contribute to \( X_0 \).

7. **Conclusion for Larger Sets:**
   - For the set \( C = \{1, 2, \ldots, 3k+1\} \), the largest class will be \( X_1 \) because the addition of \( 3k+1 \) shifts the sums of subsets of \( B \) by 1 modulo 3, increasing the count of \( X_1 \).

8. **Final Set Analysis:**
   - Since \( 2008 = 2007 + 1 = 3 \cdot 669 + 1 \), the set \( A \) is of the form \( \{1, 2, \ldots, 3k+1\} \).
   - By our previous analysis, \( X_1 \) will be the largest class for the set \( A \).

The final answer is \( \boxed{ X_1 } \) is the largest.