Siberian Forest Cats!
We have 2. Got them over a year ago. Not one reaction from my dh, who is very allergic to cats (he can step in someone's front door and know if there's a cat in the house).
I found a breeder an hour away. We spent 2 hours in her house with her 20 adult Siberian Forest cats, and many kittens. They all climbed all over dh, we sat on her furniture (which had a good amount of cat hair on it), etc. Not even the slightest reaction from dh.
The girl we picked was 14 weeks old, and groomed dh's head for an hour straight while we were sitting there. She choose him

He couldn't believe it - he could never even pet a cat before. A couple of months later, we got a little boy (who's a big brat and only tolerates us, but we love him anyway. The girl is a huge sweetie pie with everyone).
We paid about $1000 each (maybe $1200 for one). It was pay for a cat, or never have a cat, so it was worth every penny!
The breeder explained that they started importing them from Russia in the 90's, and they're breed by using blood tests to minimize the protein in their blood that people are allergic to. This protein is excreted through a cat's skin and saliva, and becomes airborne and on the fur. So each litter they test, and only breed the lowest protein ones.
One of the vets we use is from Russia, and says it's funny to her, because growing up in Russia, these cats are their 'feral' cats, and are everywhere and super-duper friendly. She was surprised how non-friendly American feral cats can be.
We're so happy with them, and I highly recommend then to everyone asking about allergies. Good luck!