Neutrogena BAR soaps are nearly scentless-- though the liquid stuff is most definitely NOT.
The one thing that I can also think of which is almost entirely scent-free (or the scent washes REALLY cleanly, anyway) most likely contains allergens for you--
Suave cucumber-melon body wash and shampoo.
DH recently came home with some stuff that I think is also VERY light in the scent department (we both have to be careful there, too)--
It's White Rain brand, lavender-vanilla body wash.
I basically use the sniff test on the cheap stuff because I got tired of spending $$$$ on mail-order only to find out that, while 'scentless' I didn't appreciate the
au de rancid animal fat scent EITHER.
I don't know if you can tolerate doing that, but a lot of the "natural" scented (well, theoretically so, anyway) stuff from Suave, White Rain, and VO-5 are pretty easy on the airways. Green tea is a scent that I tolerate very well. I think this is a very individual thing, though. I find that herbal family of scents don't bother me as much as florals or musky stuff does.
Stay far, far away from Ivory and Dove. OMG... stinky, stinky, stinky, with the possible exception of Dove's lemongrass shampoo.