But No Elephants by Jerry Smath
We found this one at goodwill and it became a fast favorite. Grandma Tildy begins the story all alone, but slowly her house fills up with different animals as a traveling pet salesman shows up and gets her to buy various animals. She always agrees to buy a pet, but firmly states that there will be "no elephants." She stays true to her word until the salesman ditches the elephant on her porch in the middle of the winter. She won't let him inside until she hears him crying. At this point, juliet likes to whimper pitifully until we turn the page and Grandma Tildy lets him in. In the end, the elephant proves himself useful to Grandma and they all live happily every after.