I noticed that the question Is Amy Pond Clara's grandmother? has a staggering eight deleted answers.
Do any of the site's questions have more than that?
Technically it's a tie between two deleted questions (10k+ only):
Both of which have 24 deleted answers. But something tells me that isn't what you meant, so:
Technically, the winner is How was Ronan able to handle the Infinity stone without being destroyed?, with 11 deleted answers. As amaranth notes in comments, however, this is a somewhat degenerate case; all 11 deleted answers spawned from a single, confused (possibly troll-y) user.
If you want to exclude that nonsense, then we have a three-way tie between:
All of which have ten deleted answers.
In total there are 8 (non-deleted) questions with more than 8 deleted answers, at time of writing:
Why could Quirrell tolerate Harry's touch at the beginning of Philosopher's Stone? is an extant question with fifteen deleted answers. That surpasses any of the examples of extant questions mentioned in the other answer.