Seriously - it depends. If you have a history of asthma before the age of 13 - and no other recurrance since then - you may not be dq'd.
You chance of getting a waiver is dependent on many factors - including when you had asthma, what triggered it, what medications you took and for how often as well as the academy to which you are applying.
All of these are impossible to ascertain on this forum.
Send an email to this guy:
Larry.Mullen@dodmerb.tma.osd.mil
He is the deputy director of dodmerb. He can give you a good idea if you have a chance at a waiver or will need one.
Be careful of blanket statements - there are so many variations and misdiagnoses of asthma you just can't make them.
If you want to give it a chance then go for it! Get your application in and your exam completed - this will give you time for dealing with a waiver....
BTW - two years ago plenty of people advised my daughter and me that she would be DQ'd for asthma and it was not waiverable. They were WRONG! She is a plebe at USMA.