If there are grandparents and/or other relatives asking "What does Susie want for Christmas," ask them if they'd be willing to get her one outfit for the doll, or one accessory. Then you get her the basic doll ensemble which usually includes the doll and a few other small items and sometimes one book. Or if one grandparent usually goes big-time on a gift, you can get that grandparent to go in with you on the doll, perhaps.
You're going to get a lot of posts today saying to buy her a knockoff doll that looks similar. Only you know if that is going to work -- she will know the difference, and if she has been seeing and playing with other kids' AG dolls she will truly know it's not AG. They are good quality and the hair and skin are definitely different from those on the cheaper dolls. Also, does she want one of the specific historical character dolls? Or a "just like me" AG doll with very specific hair, eye and skin colors? That would mean a knockoff would definitely not be what she's thinking of, if she is talking about a historical character or just like me doll.
If you want to try the knockoff route, that's fine, but talk to her about it! Discuss how if she gets a Whatever Brand Doll, she will be able to have many more outfits and more accessories, while an AG doll will mean maybe one extra outfit and that's all. (If she is expecting Santa to make it all happen -- well, you may have to have another "Santa gift" so you can actually discuss the doll with her.) She might just surprise you and say another doll is fine because she understands she can have three outfits with that doll instead of just one with a real AG doll.
As far as clothes and accessories: You can find lots of outfits online that fit these dolls. Some are just as pricey as official AG clothing but others aren't. Be aware that the official accessories can be very expensive indeed (over $100 for things like a doll bathtub or doll vanity) but would your child enjoy making accessories? My daughter made a tub and vanity and other items for her dolls from boxes, perfume bottles from beads, etc. Much better than buying plastic stuff.
The best thing about AG is not the dolls per se, but the books, especially the historical book series. Be sure someone gets her some AG books that would interest her.