I'm a relatively DM in a relatively new 5th edition game, and one of my players, let's call him E is constantly trying to be Goku. Personally I have no problem with this, but I've tried to express that this game we're running doesn't mesh well with what he has in mind, and no already existing class does what a Saiyan can do (While being balanced) and E won't make the effort to use homebrew subclasses, what would more experienced DMs do in my place?

Well, the Way of the Sun Soul subclass is arguably the closest thing you will find in D&D to saiyans

I would recommend your player give that a try, and also remind them that this is a cooperative game with many characters, so they will not reach "Goku" levels of power because their character is not going to be the "main character" of the game. 

Thanks, I'll look into the Sun Soul, the problem originally was that I didn't have the book featuring it, but I'll definitely recommend it to him.