Throughout your entire pups life they should be getting the necessary nutrients, vitamins and minerals from their food (which is why it is so important that they be on a carefully selected diet). However, dogs can certainly benefit from extra vitamins and minerals as long as they are not in excess.
For example, dogs prone to arthritis and/or hip and joint issues could certainly benefit from a joint supplement with glucosamine and chondroitin, and this can be given early in the dogs life as a preventative measure.
I also give my dogs salmon oil (fish oil) to add some vitamins to their diet and to help their skin and hair.
Overall, as long as you are giving things in moderation, and consult with your veterinarian first to ensure your pet is healthy enough for whatever you want to give them, your pet could certainly benefit from some added nutrients.
I give my poodle salmon pills to help with skin, coat, brain etc. I also give her something for joints since that breed tends to get "poodle knees"