Opera would be a good example of a third independent implementation successfully competing with the primary two (IE and NN/Mozilla). I remember switching browsers back and forth several times back then - I was an Opera 6 user, then moved to Firebird/Firefox shortly after it was released, then back to Opera 7/8/9, then to Chrome.

It was a fairly brief period, unfortunately - roughly between IE losing definitive market dominance, and Chrome acquiring the same.