I think places like Australia and New Zealand take people. I'm sure i read it somewhere. However that might of been about them poaching people from the UK armed forces because they were short. Not sure.
And as mentioned, you could always tell a few fibs. As mentioned, pilots often say they don't get hayfever etc, and if questioned, explain it as a summer cold.
How bad is your asthma? Not knowing much about the subject, how you describe it means little to me!
Mark
Oh and i wouldn't join the french foreign legion, i've looked at it and websites even explain how to go AWOL, so it can't be such an awesome unit to be in - altho it isn't full of criminals as you may think!