There may be more than one person so list the answer below if so.

Vegeta, Tien, and Trunks

Flamming Dogs Tien used the kamehameha against Roshi and Trunks used the Kamehameha wave against Bio Broly.

The only one to not use it was Vegeta. Anyway why does this matter?

Well Bio Broly was a movie, the movies are fillers as with any other anime. So wouldn't that mean that Trunks technically never had the ability?

Its not an ability to not have. Its a wave of energy that any one is capable of producing. As much as I argue about filler he still used it and that is what this question was about, who did not use.

You see me argue with Riddler and his love with Garlic Jr. saga all the time. This is because he is stating something that is not true. I accept it into the story but I'm not going to say its not filler.

You see even though Trunks may not have used the kamehameha in the real story he still could have. I look at this issue like I look at this issue the way I look at Oozaru Raditz, Nappa, and Turles being BT2 and BT3. They never transformed in the show but we verry well knew they could have and have many times in the past.

End of ramble

you mean anybody that learns how can use it but not true, androids can't use it(normal andriods not cell), unless they have some sort of knock of imatation bulit in there arms, who else can't use it?

But that wasn't the question BP. Obviously anyone could of learned it, but not everyone did. So in the actual story line Trunks didn't know how to do the kamehameha wave but in the anime story he did. Lets go with that.

BS I tell you move your hands to your side and build up energy into a small ball and unleash. Any one in the DBZ universe could do that including androids because they can shoot regular ki and why not a kamehameha.

The level at which you all take this to.

Then why wouldn't Vegeta be able to do that? You have to perform some type of special art to do the move.

Fact of the matter is he could do it.

Vegeta never cared to learn the Kamehameha-wave perhaps?

Originally it would take 50 years to learn Kamehameha(stated by Master Roshi that it took around 50 years) but the characters turned out be more super natural than the other strange things.

Actauly Trunks did the kamehamehawave in GT at least twice so that means he could have learned it. And just because its filler doesent mean its not real so Trunks still coulda learned it in DBZ. And I agree with BP on this matter. There are a lot of characters whom never learned it TZF but naming them all would take a long time. I could name a few though: the Androids minus Cell, Vegeta, Saibamen, Dodoria and Zarbon, Freiza and his men, Kami, Mr. Po-Po, General Tao everyone in the red ribbion army, Garlic Jr and his men, Broly, Cooler, Bojack, Turtles, Lord Slug, Choutzu, Yajarobie most of the characters in dragonball, General Rildo, the sigma force, the Ginyu force members, Piccolo, King Piccolo, Bardock, Nappa, Raditz, the shadow dragons, Baby, Chi-Chi, Videl, Hercule, and there are obiously others but that is a lot as it is.

Naming characters isn't the real point.

Tien does learn the Kamehameha in Dragon ball during the World martial arts tournament.

Vegeta is the only character above that never bothered to learn the move.

Many others could've learned it if they felt it desirable.

And I know that whoever marked Goku had to be kidding,right?

Well Goku never did copy the attack correctly. Looking at it like that the only ones to learn it were Rohshi and Grandpa Gohan and Tien.

Several others use it but without the original and movements.(in the Deadzone movie I thought that I saw Goku use it correctly)

Yamcha taught himself the move but doesn't do it the original way either.