We have seen a real email address without any dot, and it routes successfully to a TLD MX exactly like you describe so yes, it does happen. :)
But yeah, that's effectively dead and you probably don't want to try sending mail that way.
Edit: punctuation.
There's very good odds that the email they send will have their "From:" (or "Reply-To:") address correctly set. Then just have an email autoresponder which emails them back a link with a token in it, when they click on that it'll take them to a page to create their account, with their email address already filled in by the token.
Ideally, all this would be done away with by OpenID or client-side certificates or something along those lines. The whole password-and-email-in-case-you-forget-it paradigm is broken, really.
A simple check for an "@" sign would go a long way to avoiding bounced mail notifications from usernames being entered in email address fields.
But not all the way. And so a simple "you might have got this wrong" flag would be helpful, no?
In our case, accounts may be created by third-party apps using REST JSON APIs, so we want to let the third-party app know that the email address isn't RFC-valid.
Of course, whether any email server actually accepts that is a different story.
Anyway, the regexp I use is /.+@.+\..+/ which supports the address you describe, but (usually) catches the relatively common mistake of user@yahoo,com
So on our production servers, I need the @, but on our dev/text servers, I don't. And no, the domain is not appended to the address before sending. That's the part I actually know about.