In order for this quadratic to have at least one real solution, its discriminant must be non-negative. In other words, $ b^2 - 4ac = 5^2 - 4(2)(c) = 25 - 8c \ge 0 $. Rearranging, we have $ 25 \ge 8c $. Dividing by 8, we have $ 25/8 \ge c $. Therefore, the largest possible value of $ c $ such that this quadratic has a real solution is $ \boxed{\frac{25}{8}} $.