Let the two consecutive integers be \( n \) and \( n + 1 \). The sum of these integers can be expressed as:
\[ n + (n + 1) = 2n + 1 \]
According to the problem, this sum is no more than 209:
\[ 2n + 1 \leq 209 \]
To solve for \( n \), first subtract 1 from both sides:
\[ 2n \leq 208 \]
Next, divide both sides by 2:
\[ n \leq 104 \]
The largest integer \( n \) that satisfies this inequality is 104. The consecutive integer following \( n \) is:
\[ n + 1 = 104 + 1 = 105 \]
Thus, the largest possible integer among the two consecutive integers is:
\[ \boxed{105} \]