Ask HN: My signup form is being bombarded by bots. What do I do?

My sign up form is being filled by automated bots/scripts that are signing up using other people's emails. This means that the confirmation email my app sends out get flagged by the email owners as spam, hurting my domain reputation and email sending limits.

So far, I've tried Google Recaptcha which has reduced the rate but has not eliminated it, even at the maximum filtering level. The rate of fake sign ups is still high enough to cause email reputation problems for me. What do I do?