My 88 year-old mother made the decision not to get them any more.  She's very clear in her mind that she wouldn't have radiation or chemo if she did get cancer, so why bother?  I'm thinking that if a mammogram found a lump, she might be able to have just surgery, and do fine and live another five or ten years than if she didn't have the mammogram.

For the elderly, I think the determining factor in whether to get a mammogram is the question of what you would do if a cancer were found.  Some healthy folks in the 85+ crowd would want to fight it with every possible medical means.  Others would feel that they don't want to prolong their lives with debilitating medical procedures.  If your mom's Alzheimer's is still in early enough stages that you can have this conversation with her, then you'll have some guidelines on what would be best for her.