The first person we see reach Super Saiyan is Goku during his long battle with Frieza. At this time it seemed very clear who the legendary Super Saiyan was and if Akira Toriyama (the creator of Dragonball) had gotten his way to stop creating the DB, Goku would have definetly been the legend. However, he is not because as we all know the story did go on.
Next we see Vegeta become Super Saiyan so surely he must have been of what the legend speaks. His power was incredibly high when he finally completed the transformation and he has the royal bloodline. Vegeta also continues to train his ENTIRE life never giving up. However, he is not the legend and is just another Super Saiyan.
How about Future Trunks? He was an awesomely powerful half-Saiyan, and fought off the evil Android's when he was the only one left. Would this make him the legendary hero? Nope. Just like regular Trunks he pretty much gives up training/fighting and goes to work at Capsule Corp.
Now we come to Goten and Trunks. They become Super Saiyans younger than anyother characters with great power but, were only able to cause their fathers did. Because if you have already become SSJ and you have a child something in the Saiyan genes changes and it becomes much easier to become a Super Saiyan.
If Dragonball had ended after the fight with Cell Gohan would have definetly been the Legendary Super Saiyan. However, after he reaches SSJ2 he becomes Mystic which is a completly different thing that in no way can be called Super Saiyan, thus making it impossible for him to be the legend.
With all those warriors out of the equation, then it is all narrowed down to the final super Saiyan, the one believed by most to be the legend. His name is of course, Broly. He had strength far higher than anyone else's, and he was unsurpassable. He was destined to become a spectacular fighter with unimaginable strength, and perfectly carried out this prophecy. Even FUNimation considers Broly to be the legend. He isn't the legendary super Saiyan though.
Okay, so who the heck is? Well the real legendary super Saiyan was the one from way back that Vegeta talked about, and who only had the ability to maintain his power during a full moon. While his power wasn't as high as the other SSJ's, he was the first one known in history, and stories were told about him, passed down along the generations, thus making him a legend.
This may seem unexpected and you may even refuse to believe that someone who didn't feature in the series can be the legend. If that's the case then here's the other answer: There isn't one. Only one legendary super Saiyan has ever been born, and it wasn't Goku or his friends, but that nameless Oozaru with insane power.
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