There is no limit on the number of pets military families can own. There is a limit on how much the Department of Defense will reimburse you. You can get up to $500 back for one pet within the continental United States and $2,000 back for one pet everywhere else in the world.
If you are traveling to or from certain high-risk countries, the Defense Department has authorized reimbursement of up to $4,000. This policy is in response to the Centers for Disease Control and Prevention (CDC) implementing a regulation that requires a higher screening level for dogs returning to the United States from countries that pose a high risk of rabies.