That we know of. Robert supposedly had tons of bastards, we don't know if they killed all of them or even if they knew of all of them when that purge happened.
In books there was another bastard that Robert actually acknowledged as his own, but his story was removed, and Gendry has basically absorbed his role, ie Melinsandre wanting him.
In terms of claim to throne, technically unacknowledged bastards have no legit rights to throne, especially with other family members around, like Stannis and Renly previously.