Based on the thread title, I thought that was what he did ask.
Yes. The Rutles, some asshole pretending to be Kate Bush, Peter,Paul and not Mary, Eddie’s father, the indian who cried when he saw people littering, Billy Carter, Bewitched, wait…what are we talking about?
You can like a famous person’s page. It looks like a regular page but its actually a fan page. Not the same as a personal page for friends and family. George Takei has over 4,000,000 fans on his page. You probably know the difference but a lot of people don’t make the distinction and say they are “friends” with George. You can see in this thread some people made that mistake too.
Two, I guess. One is a magician who is moderately well-known and one is a TV presenter of a very well-known show. I haven’t added anyone just because they’re famous.
A couple of others have written books or appeared on TV in somewhat-recognisable roles but wouldn’t qualify as famous IMHO.
In my case it’s because their Facebook pages are private, ordinary facebook pages, not there for publicity. The TV presenter even uses a different name.
A couple of published authors, one of which I’m familiar enough with that I’ve sent him and his daughter presents and he’d recognize my name. The other wouldn’t know me from Adam’s off ox, but he accepted my friend request.
I don’t do Facebook (or Twitter), but I figure that over the number of years the Straight Dope has been around, it’s even money that more than one famous politician/celebrity/‘other’ from somewhere in the world has been or is an otherwise anonymous member here.
Hello, your Royal Magnificence! 
Mark Balelo was on Storage Wars and we became friends before the show got really big. He was a very nice guy, good to his fans, and he gave me a lot of business advice in private messages that HE initiated. He even invited me to meet him and tour his operation if I was ever in southern California.
Earlier this year, he was arrested for possession of meth. A few days later he killed himself. What a freaking waste. He was a kind, generous and gentle man. 
ETA: He genuinely cared about people, and didn’t post just to get his ego stroked, like most “reality” stars. Sucks.
About a half dozen NHL players, past and current. Including a few I knew IRL.
Matthew Bracken (author)
Jocelyn Tanis (actress)
My most famous FB friend is an asset manager who appears fairly regularly on CNBC and Bloomberg and has a successful blog. We share interests, I’m not a customer (unfortunately).
I have one very well known biographer. A couple of successful but not so well known TV producers and a few actors who are far from being stars.
Christy Smith—know her personally.
Carla Garapedian—we’ve never met, but she has worked very extensively with my aunt’s second husband, Mike Hagopian.
I grew up with Grammy-winning jazz band leader and film composer Gordon Goodwin.
Off the top of my head:
SJ Tucker (aka Sooj): musician
Michael Coorlim: steampunk author
Both have pretty niche audiences (but both are excellent and I recommend them).
I’m also friends with the sister of Dale Levitski. And it was years before I even knew she had a brother.
There are many others, depending on how you define semi-famous; I know a shit-ton of actors, many of whom have worked in things you’ve seen. I know the guy who played Ellis Hicks (the evil alderman’s right-hand-man) on The Chicago Code. I also know a guy who was beat up by Westen in Burn Notice. Having a “check me out on the next episode of [show]” status pop up in my newsfeed is a semi-regular occurrence.
A couple of people who are well-known in the Free State Project/Liberty community. Between them they have about 10-15 podcasts, websites, and radio shows.
A few jockeys and a few trainers. Mainly ones that generally race somewhat local - either Hawthorn or Arlington. There are one or two that don’t race here anymore (moved to better tracks) but we’ve kept in contact.
Beth Grant?
I have David Gerrold and A.C. Crispin. Those two actually granted a Friend request. I’m following others, but that doesn’t count for this thread.
A few people.
Nick Lowe’s producer
A really tall, dark and gruesome Hammer horror film star (I don’t think it’s actually him, but from what’s posted, I’d say it’s someone damned close to him)
Tim Lucas - Video Watchdog (I’m a long-time subscriber)
Monte Hellman
Ian Gomm
the daughter of Leo Gordon - one of my favorite film heavies
A.C. Lyles
Timothy Carey’s son - Romeo
Jackie Joseph
some of the folks who appeared in the “Beach Blanket” films of the 60s
other film folk
I’m obvious a huge film fan and I find that if you message politely with a "I’m normal, and I’ve admired your work especially “XYZ”, then you’re more than likely to either get a nice message in return and/or a friending.
I’ve had online interactions with most, and I’ve met Nick Lowe’s producer a number of times at his shows.
Fun stuff.
BA~