Depends . Many physicians label a lot of wheezing processes in infancy as asthma that may not be true asthma.Various lower respiratory viruses like RSV can bring out a transient wheezing tendency that disappears with age.Up to 75% of wheezy kids will not wheeze after 4-5y. This overlaps true asthma, the worst cases may present < 1y in males <4y in females & persist through life.
Outgrow asthma? Asthma is a chronic inflammatory condition in response to allergens & usually as one grows the diameter of airways also grows. One may not necessarily be outgrowing the asthma but as the airway get wider, the degree of obstruction decreases resulting in lesser degree of symptoms&at one point the person may not realize that he/she is wheezing. So some children seem to outgrow it.
Asthma... Asthma is a chronic disease that has no cure. But, some asthmatics have prolonged asymptomatic periods between exacerbations, sometimes lasting many years. However, there can be an acute exacerbation at any time. So, you can have no exacerbations once you are an adult , only to suffer an exacerbation later in life.