The swan is generally considered to be the bird with the most feathers, with a specific type of tundra swan boasting more than 25,000 individual feathers. This is a lot compared with the average for birds, which is closer to just 1,000 to 2,000 feathers per bird.
Size does factor into this question. Swans tend to be very large, so while they have the most feathers, they might not be the most densely feathered. The tiny hummingbird, which is just a few inches in size, can have almost 1,000 feathers, meaning it is a very densely feathered bird, considering its miniature stature.