Can you appeal a play twice?

Answer 1

It all depends on how calmly you appealed the play to begin with. If you were professional enough and didn't get thrown out of the game, then you just might have the opportunity to appeal it again. However, the chances of your getting tossed with the second appeal will go up exponentially.

Answer

The answer above is refering to similar plays in the same game i believe. But the way i understood your question was asking if you can appeal and appeal essientially. the answer is NO. You can appeal a call and if you still do not agree with the call and you feel it is a misrepresentation of the rule you can then "Protest" the call and finish the game under "protest". The League office or in MLB, the Commish office, will then review the play with the umpires and make a ruling, if they feel it had an effect on the outcome of the game then they can order the game to be replayed