Technically Gable wasn't allowed to fly combat missions, he was there as a gunnery instructor and to watch them over England as they practiced gunnery. But he sort of sneaked on a number of true combat missions as "observer" until higher ups intervened to stop him.
And one other that reportedly tried to enlist but was denied was John Wayne. It was said he felt guilty seeing so many Hollywood folks, known and unknown, enter the service. He tried to enlist but was rejected, supposedly because the head of Republic pictures was a friend of the Secretary of War and requested that if Wayne tried, he should be rejected since Republic didn't want to lose him like so many others when making films during the war.