I don’t know about growing out of allergies, I’d imagine it’s possible; I seem to have grown INTO an allergy.
A few years ago, on a few occasions, I suddenly got sick for seemingly no reason. After about the third time, I realized that each time I had eaten fish. A while later I tried to get scientific about it, set aside a night when I had nothing to do, and ate some fish. It was horrible–throwing up every 15 minutes, my temperature dropped really low; it went on for hours.
I loved fish and really miss it now. I told myself at the time, “Never Again!”, but one of these days I may actually make another attempt, just to see if anything’s changed.