This is one of the three records listed in the Guiness Book of Records as having been recorded in more than 1000 versions. The others are Yesterday and Tie a Yelow Ribbon.
We thought your question was so good that, believe it or not, we emailed Paul Anka, who wrote the original, to try to find out. Even he doesn't know the exact figure because the list just goes on and one, but he says it is "something in the region of 350". That really does comes from the horse's mouth, so if it doesn't get a three star rating, what will?