Does the book ever mention how many times Atticus had appealed the case of Tom Robinson?

Atticus was never able to file an appeal because that can only happen after a trial when one party is not satisfied with the ruling. Just days after the Tom Robinson trial ends, Tom dies before Atticus can file the appeal. In addition, Atticus told Tom that they would probably lose at first, but he had high hopes saving Tom in the appeals process. Atticus mentions his plans to his brother Jack in the following...

Atticus was never able to file an appeal because that can only happen after a trial when one party is not satisfied with the ruling. Just days after the Tom Robinson trial ends, Tom dies before Atticus can file the appeal. In addition, Atticus told Tom that they would probably lose at first, but he had high hopes saving Tom in the appeals process. Atticus mentions his plans to his brother Jack in the following passage:

"Before I'm through, I intend to jar the jury a bit--I think we'll have a reasonable chance on appeal, though. I really can't tell at this stage, Jack. . . You know what's going to happen as well as I do, Jack. . . Why reasonable people go stark raving mad when anything involving a Negro comes up, is something I don't pretend to understand" (88).

Atticus says this to his brother at Christmas time, about six or more months before the trial. First he states that he hopes to have a reasonable chance at an appeal and that he and Jack both know "what's going to happen." That means they know that no white jury in the South will ever side with a black man in a case against a white. After Tom dies, Atticus says the following:

"We had such a good chance. . . I told him what I thought, but I couldn't in truth say that we had more than a good chance. I guess Tom was tired of white men's chances and preferred to take his own" (235-236).

In summary, there was no appeal ever filed because the trial had to happen first. The evidence provided above backs that up as well as revealing the racist nature of the Southern justice system during the 1930s. Atticus felt better about the appeal after the trial, too, because the jury stayed out longer than he had expected, which meant that the all-white jury actually debated the issue rather than simply convicting Tom in 5 minutes. Unfortunately, Tom didn't see it that way.