My humble answer: During an acute exacerbation of asthma you will have air trapped in your chest, and this can give you a barrel chest.
What is a barrel chest: The mayo clinic defines it best: The term describes a rounded, bulging chest that resembles the shape of a barrel. Barrel chest isn't a disease, but it may indicate an underlying condition.
Is a barrel chest permanent: When I was a kid I often was told I had a barrel chest, but I do not have one now. For asthmatics it's only temporary until the asthmatic attack is resolved and the excess air is allowed to escape from your chest. In this way, part of the definition of asthma is that it is completely reversible, either with time or medication. However, patients with emphysema may have permanent lung destruction that results in air being permanently trapped in the lungs. This may result in a permanent barrel chest.