Does each team have a limited number of appeals to the third umpire?

Yes, in cricket, each team has a limited number of appeals to the third umpire, typically one unsuccessful review per innings for on-field decisions. If a team's review is successful, they retain their right to review another decision. The system is designed to ensure that teams use their appeals judiciously. However, some decisions, such as run-outs, can be referred without using a team's review.