TL;DR: Ser Robert Strong is the reanimated corpse of Ser Gregor Clegane, the Mountain.
Gregor Clegane — “the Mountain” — supposedly died when dueling Oberyn Martell to decide Tyrion Lannister’s fate. The Mountain, as his name suggests, is a massive man. He was a ruthless killer and was nigh unstoppable in one-on-one battles. Martell defeated him (sort of; Martell died too) with the help of a poisoned spear.
“My champion will need a new name as well as a new face.”
-A Dance With Dragons, Cersei I
So what makes us think that The Mountain got the Frankenstein treatment? Because of the man who tended to the wounds which ultimately killed him: Qyburn.
Ser Robert Strong is not talked about too much in the A Song of Ice and Fire series, but when he is discussed, the links between he and the Mountain are easy to spot. Robert Strong is massive — just like the Mountain — and his origins line up with the Moutain’s death.
This theory is almost canon; it is widely accepted to be true, and we’re merely waiting for the confirmation. (You never really know with GRRM.)
Qyburn is a former Maester who lost his chain for conducting seedy experiments on humans, including The Mountain. In A Feast for Crows, Qyburn says he “opened the bodies of the living” so that he could understand death, and that’s why his chain was taken.
Queen Cersei then gives Qyburn permission to open up the Mountain’s dying body, and asks only that Qyburn give her the Mountain’s head when he finally died. She was going to ship it to Dorne to uphold her father’s promise.
We later learn that the Mountain did indeed die, and that his head was severed from the body, cleaned, and prepared so it could be shipped to Dorne. It’s important to note that only his head was shipped to Dorne, and we don’t know exactly what happened with his body.
And from there, we can only speculate.
We only hear of Ser Robert Strong once the Mountain has passed. It’s also clear that Ser Robert Strong is a concoction of Qyburn’s created to serve as Cersei’s champion. The Mountain served as her champion before, so it would make sense for her to trust the largest man in the land to defend her once again.
This undead Gregor — unGregor, as some have called him — might literally not have a head, which would need to be the case for the “Ser Robert Strong is the Mountain” theory to be true.
“Tell me, ser, where did this man come from?” demanded Mace Tyrell. “Why have we never heard his name before? He does not speak, he will not show his face, he is never seen without his armor. Do we know for a certainty that he is even a knight?”
We do not even know if he’s alive. Meryn Trant claimed that Strong took neither food nor drink, and Boros Blount went so far as to say he had never seen the man use the privy. Why should he? Dead men do not shit. Kevan Lannister had a strong suspicion of just who this Ser Robert really was beneath that gleaming white armor. A suspicion that Mace Tyrell and Randyll Tarly no doubt shared. Whatever the face hidden behind Strong’s helm, it must remain hidden for now. The silent giant was his niece’s only hope. And pray that he is as formidable as he appears.
-A Dance With Dragons, Epilogue
He is never seen without his armor (including his helmet, because he may have no head). He doesn’t show his face (again, because he might not actually have one). He does not use the restroom, or eat or drink, according to what people have seen. “We do not even know if he’s alive,” the quote says.
All signs point to Ser Robert Strong being dead, and all signs say that Ser Robert Strong is the Mountain; or at least the Mountain’s body equipped with some type of head.