Neptune is not the largest planet in the solar system, but it does have the largest Hill sphere. Crudely speaking, this is the sphere in which a planet is the dominate gravitational player.

It is usually said that a prograde satellite will only be stable if its orbital radius is less than half the Hill sphere radius. For retrograde satellites, less than two thirds. Different sources give different fractions, it is a bit fuzzy, but that’s about right.

The two satellites with the biggest orbital radius are Neso and Psamanthe. They are both highly eccentric retrograde satellites of Neptune, further (on average) from Neptune than Mercury is from the sun.

Psamanthe:
Average orbital distance: 47 million km (minimum 26 million km, maximum 68 million km)
Orbital period: 25 years
Diameter: ~40 km

Neso:
Average orbital distance: 48 million km (minimum 24 million km, maximum 72 million km)
Orbital period: 27 years
Diameter: ~60 km

These diameter estimates are very rough, as they are based upon albedo guesses.

You may find this interesting as well.

“As it orbits the Sun, (469219) 2016 HO3 appears to circle around Earth as well. The object is beyond the Hill sphere of Earth, meaning that the Sun exerts a stronger pull on it than Earth does. Although it is too distant to be considered a true natural satellite of Earth, it is the best and most stable example to date of a near-Earth companion, or quasi-satellite.”

Swirly thing has gone South….

https://earth.nullschool.net/#current/wind/surface/level/orthographic=-230.54,-28.55,736

Sorry, forgot where I was.