I know some cases of a third set of teeth and even one case of a fourth set have been recorded in humans, but these are not the norm. 

Elephants go through six sets of teeth in their lifetime, but humans only two. We have more or less the same lifespan as elephants, so why don't we get six sets too?

In modern humans, tooth wear is not really a problem as most of our diet is relatively soft. Elephants on the the other hand have to eat an awful lot of food because of how big they are, and the food they eat is often very tough to chew (twigs, branches, tree bark, roots as well as grasses, bushes etc). This wears their teeth very quickly, so they need these replacements. However, the elephant doesn't really have sets of teeth -instead each new tooth appears at the back of its mouth, and pushes the old, worn front tooth out of the front of the mouth, breaking off and falling out bit by bit! In this way, the elephant only really has one enormous tooth working at any one time plus the broken older one being pushed out. This has been compared to a conveyor belt, pushing forward the new teeth and pushing out the old, worn ones.

A couple of other mammals have this type of tooth replacement - the swimming manatee, which is related to the elephant, a species of kangaroo, and possibly an african mole rat! Most other mammals have tooth replacement more like humans, but lots of rodents only ever have one set of teeth, and there are lots of other weird patterns of tooth replacement across all mammals!