Why does Son Goku at Dragon Ball not ask the Dragon to be immortal?

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.

I haven't actually watched Dragon Ball Z, but based on my knowledge of it, there are a few possible reasons:

• The Dragon Ball wishes are commonly used by the heroes to revive the recently-dead (an example being Krillin). So not only would wishing for immortality be extremely selfish of Goku, and disrepectful to his friends who died in battle, but it would also be pointless because he can always be wished back to life himself (and I believe he is on at least one occasion).
• Goku is not Frieza. He only wants power so he can protect those he cares about, not just for its own sake.
• Goku is not the smartest person. Such a wish might simply not have occurred to him.
• It would make things really really boring. The show would just be Goku defeating every villain with ease because he can't be killed, and it would remove all the tension out of the fight scenes because you know Goku is going to win.

Goku doesn't want to be a hero in any situation and he doesn't want immortality. He always wants fights with strong enemies more than saving the world. Becoming immortal will ruin the feeling of his thirst for beating strong enemies as a martial artist. It's made clear in the original Dragon Ball series.

In episode 146, when Goku is fighting with Ma-Junior (Piccolo Daimao), Kami-Sama comes to rescue Goku from one of the strong punches of Piccolo. He also tells Goku that they can both fight together against Piccolo, so that Goku can save the world by eliminating the Piccolo Daimao. But Goku stubbornly refuses his help and asks to punch him with the same power which is stopped by Kami-sama. At that time, Master Roshi mentions the real nature of Goku:

"It would seem the world isn't particularly important in his (Goku) eyes. Beating opponents who are strong enough to excite him is all he wants."

While I agree that the better answer is that Goku is more interested in a loyal fight against the strongest warriors, and has no desire to live forever, I propose another possibility.

[Warning: some spoilers on the plot of Dragon Ball Z]

Goku has passed a lot of time in the afterlife in DBZ.
Really, being dead is not an issue for the strongest characters in Dragon Ball: Goku could train with King Kai and also fight against some of the strongest warriors in the history.
In some particular situations, some characters were also allowed to come back to life (Goku to attend a tournament, Vegeta in order to fight against Majinbu). The fact that death is not an issue was also sometimes mocked in the anime itself, like when the Elder Kaiohshin sacrifices himself to resurrect Goku: 2 seconds after he's standing again, only with the halo...

So, Goku has no fear of the afterlife, since it is clear that in the unoverse of DBZ death itself is not a frightening concept