I will base my answer by Dragon Ball Super where Toriyama finally got Goku animated the way he always intended. People who watched Dragon Ball Z and Super may notice that Goku has a really different personality because of the conflict he had with the animators, as seen in this interview.
When referring to the "Poison" in the comics:
Wired: There’s actually “poison” inside?
Toriyama: Right. There’s how, basically, Son Goku from Dragon Ball doesn’t fight for the sake of others, but because he wants to fight
against strong guys. So once Dragon Ball got animated, at any rate,
I’ve always been dissatisfied with the “righteous hero”-type portrayal
they gave him. I guess I couldn’t quite get them to grasp the elements
of “poison” that slip in and out of sight among the shadows.
So, basicly, Goku is not supposed to and anyone who got to watch Dragon Ball Super will notice that: Goku is not a hero. At many points in the series, Goku almost wrecked the Universe because of his selfish acts as he is constantly looking for a fight with stronger and stronger characters.
By this reasoning and watching the show, it is plausible to assume that Goku will never wish for immortality. That is because Goku's main reason to live is to get stronger and to fight tougher enemies. However, he will never cheat to the point where he becomes immortal because that would ruin the fun of fighting. He lives to be challenged.
There is a point in the story where a character called Zamasu reveals to have become an immortal through the power of the Super Dragon Balls. I don't recall perfectly, but I think Goku considers him a cheater.