jmc

We use the Euclidean Algorithm. $$\begin{array}{rl}\gcd (11n+3,6n+1)& =\gcd (6n+1,(11n+3)-(6n+1))\\ & =\gcd (6n+1,5n+2)\\ & =\gcd (5n+2,(6n+1)-(5n+2))\\ & =\gcd (5n+2,n-1)\\ & =\gcd (n-1,(5n+2)-5(n-1))\\ & =\gcd (n-1,7).\end{array}$$Therefore, if $n-1$ is a multiple of 7, then the greatest common divisor of $11n+3$ and $6n+1$ is 7. Otherwise, the greatest common divisor is 1. This implies that the maximum possible value for the greatest common divisor of $11n+3$ and $6n+1$ is $\boxed{7}$.