Twins happen in one of three ways. One way is that two eggs are released and they are each fertilized. The next way is that one egg is released and fertilized by two sperm. (Very rare 1% of twins and they usually don't survive). These are fraternal twins. (Although the second kind are called semi-identical because they have the same egg.)
The other way is when a fertilized egg splits in two. This make identical twins. If the egg doesn't split completely, then you have conjoined twins.
What you are talking about would be triplets. One fertilized egg would split for identical twins and another egg would be fertilized. And yes, it has happened. But often, there is no way to tell if they all look alike unless they have DNA testing.
For higher order multiples, it is more common. The Fischer quints, born in the 60s, had 3 identicals (one egg split three ways) and 2 fraternals. ]
I think what you are saying is that a set of twins gets created by two sperm hitting one egg, then splitting into two as semi-identical twins, then one of those splitting into two as identical twins, resulting in triplets - two identical and the third semi-identical to the two. I couldn't find any examples of that, but it would take DNA analysis to prove it and most folks dont' get that. So while it is possible (and very rare), there aren't any documented cases of it that I could find. Of course, they could have happened and the testing just hasn't been done.
Pink, go back to nursing school. You failed this one. Which makes me wonder if you are lying about being a labor and delivery nurse.