We want to find the maximum possible value of $ \text{gcd}\,(F_{n}, F_{n-1}) $. Since $ F_{n} = F_{n-1} + F_{n-2}, $ by the Euclidean algorithm, this is equivalent to finding \begin{align*} \text{gcd}\,(F_{n-1} + F_{n-2}, F_{n-1}) &= \text{gcd}\,(F_{n-1} + F_{n-2} - F_{n-1}, F_{n-1}) \\ &= \text{gcd}\,(F_{n-1}, F_{n-2}).\end{align*} It follows that \begin{align*} \text{gcd}\,(F_n, F_{n-1}) &= \text{gcd}\,(F_{n-1}, F_{n-2})\\ &= \cdots = \text{gcd}\,(F_2, F_1)\\ &= \text{gcd}\,(1,1)\\ &= \boxed{1}.\end{align*}