From what I understand, and I am by no means and expert, they will probably request a remedial (more paperwork) and/or order a test such as a speriometry or a methacholine challenge test. Unless you are currently being prescribed medication (which can pose a challenge, even if you never use the medication), I do not see why they would disqualify you, especially if they have you do these tests. However, all exams are considered on a case-by-case basis, which means they still could disqualify you. If they do, do not despair! If the rest of your application to the service academies or ROTC is competitive, then I would say that you will be offered a waiver by the school.
Again I cannot speculate, but if your asthma history completely ended before you were thirteen I believe you will be fine.
Hope this helps and sorry for the long post.