jmc

Every five-digit palindrome is of the form $ABCBA$, where $A$, $B$, and $C$ are digits. A number is divisible by 6 if and only if it is divisible by both 2 and 3.

The number $ABCBA$ is divisible by 2 if and only if the digit $A$ is even, so the largest possible digit $A$ is 8. The number $ABCBA$ is divisible by 3 if and only if the sum of its digits, which is $2A+2B+C$, is divisible by 3.

The largest possible digit $B$ is 9, and if $A=8$, then $2A+2B+C=C+34$. The largest digit $C$ for which $C+34$ is divisible by 3 is $C=8$. Therefore, the largest five-digit palindrome that is divisible by 6 is $\boxed{89898}$.