jmc

The largest possible 4-digit palindrome in base 3 is $2222_3=80_{10}$. We know that converting $80_{10}$ to other bases will result in 3 digits when the base is from 5 to 8, inclusive, since $4^3<80<9^2$, meaning $1000_4<80<100_9$. Converting to other bases, we get $310_5,212_6,143_7,120_8$. The only palindrome is $212_6$, which is expressed in base $\boxed{6}$.