Like many planets, Neptune’s orbit isn’t exactly circular. Instead, Neptune orbits the Sun in an elliptical orbit. At its closest point, Neptune gets within 4.45 billion km, and then orbits out to a distance of 4.55 billion km. It takes almost 165 years to complete one orbit around the Sun.
It’s a shame Pluto isn’t a planet any more, because it’s really far. Pluto gets as close as 4.44 billion km. But its orbit is so elliptical that it gets out to a distance of 7.38 billion km. In fact, there are times in Pluto’s orbit when Neptune passes it. Then Neptune really is the farthest planet from the Sun, whether or not you think Pluto is a planet.
What’s farthest object from the Sun? Astronomers think that the long period comets come from a region of the Solar System known as the Oort cloud. It’s possible that this region extends out from the Sun to a distance of 50,000 astronomical units (1 AU is the distance from the Earth to the Sun).
Here’s an article that lists the distances to all the planets.
And here’s an article from Solar Views that talks about the Oort Cloud.
We have recorded an episode of Astronomy Cast just about the Sun called The Sun, Spots and All.