Dogs, like people, only have two sets of teeth. Dogs are born toothless; their milk teeth begin to come in when they are between 3 and 6 weeks of age. Their adult teeth do not come in until they are at least 4 months old.
Dogs and people are both diphyodonts -- animals that only grow two sets of teeth throughout their entire lives. Mammals in general are diphyodonts, but most other kinds of animals are not. Sharks famously grow new teeth throughout the length of their lives, but this fearsome fact is actually true of most fish. Similarly, crocodiles and alligators are able to grow new teeth as they lose them, but so can geckos.