is there anywhere I can get a free .me domain name?
Always buy your domain name. The cheapest ones start from 2 dollars up to 15... it's worth it. I once used free co.cc domain for my project and when it got bigger, I troubled so much moving the domain. Usually free domains are used by spammers so people/websites don't trust them either. For an example you can't post any domain ending with co.cc on Facebook. Even though my site is legit, I still can't share it on Facebook.
When it comes to domains in general it's always wise to buy the domain yourself rather than use the freebie bundled one because essentially with free names, the hosting company owns the name and if you switch hosts and cancel the account, your prior host will still have the domain.
Going back to the .me question however, I have never seen a web host or registrar offer a .me for free - and this is mainly because as it is an international domain, the country owning the domain controls the prices whereas with .coms and such, they're a bit more competetive price wise.
Still you can typically score .me domains at discounted rates from many of the top registrars. I once had a few .me's at $10 a pop awhile back although $15-$20 seem to be the going now.
My favorite registrar for international/vanity domain endings is gandi.net since they offer over 90 domains and the prices are very reasonable.