The Sun's is a fairly ordinary star. Although somewhat larger than most within our galaxy, the Sun's main claim to fame, that which makes it most extraordinary, is its proximity to Earth, it's our star and the fundamental building block of our solar system.
It was only within the last hundred years that the mechanism by which a star could 'burn' for more than a billion years became understood. The fusion reaction which maintains a stars energy production is proportional to its mass with more massive stars living shorter more energetic lives.
The main sequence stage of our Sun is expected to last about 9.5 billion years and it is currently at about mid-life. For a star to have a productive life of greater than the current 13.7 billion years of our universe it would have to begin its life as a much smaller star. With less mass and therefore less density and pressure at its core, a black dwarf can sustain the fusion process for a much longer duration than a larger more massive star.
By the way, the fusion process within very large stars which live short and energetic lives ending in a cataclysmic explosion, is where the heavier chemical elements which gave birth to our solar system and subsequently ourselves were created.
http://nineplanets.org/sol.html
https://www.youtube.com/watch?v=JN4nF4kUNpg