Certain allergies. From what I've heard anecdotally, most medication allergies are OK. Hay fever and the like probably isn't an issue. Food allergies are more problematic but it probably depends on the type of allergy (what you're allergic to) and the severity of the response.
I suggest checking with DODMERB for details and definitive info.
In my case, I used to have food allergies that I grew out of and I have the occasional air allergy attack, but I was deemed fully qualified by DoDMERB. Take this with a grain of salt however, as every case is unique.