Youtube – Can a user still see their video on their account if YouTube has removed it

According to threads in the YouTube's help forums, the copyright strikes are permanent and are not removed without filing a counter-notification".

Excerpt from the page:

If you believe your content was misidentified as infringing, 
you may file a counter-notification. If you did not have all
of the rights to post the material at issue, you MUST NOT 
submit a counter-notification. 

Good luck!

Full Disclosure: I am the lead developer on the site I'm mentioning...

If you can get the URL to the .FLV file (just google 'how to get FLV URL youtube'), Upload by Transfer is what we call it. That feature gets released next week on our platform (see blog post for details). You provide the URL to the FLV, and the cloud service pulls it off and transcodes it.

Edit: Just tried this with the KeepVid suggestion in another answer. Just picked a popular YouTube video... http://play.nimbushd.com/view/lfsv

This is normally meant for production shops with automated workflows (HTTP/FTP servers) but FLV is supported and it's no different. And in the interest of neutrality I suppose you could also do this with encoding.com if you had a destination for it (an FTP server or something).

If you're not looking for a cloud service/hosted platform, then of course there are YouTube downloaders and you can just save the .FLV files off to your hard drive.