Nothing is cheap in Asheville but both the Grove Park and the Biltmore are to be thought of as experiences. We've never stayed at the GP but have eaten dinner there which was wonderful. We celebrated a big anniversary a number of years ago at the Biltmore and the ambiance and the food was worth it - once. We had a ground level terrace room that looked out over a grassy field and a wedding was being held there. The bride, who was of Indian descent, wore a flowing, brilliant red sari embellished with many, many diamonds. She actually had body guards surrounding her trying to be discreet. I don't know who she was but she must have been the daughter of someone very important.
The arboretum is beautiful and there is a smaller botanical garden closer to downtown that has mostly native plants.