I have only my own experience to base this advice on, which is meant to show you a few super-casual ways to meet someone famous. Whether you talk to them or not is up to you. Whether you pull up your shirt and ask them to sign your boobs: also up to you. I take no responsibility for how you use this advice, I'm just telling you what has worked for me.
1. Live in a large city. Large cities are hubs of industry, finance, and entertainment. Let's say you want to meet famous actors. Where do successful actors live? New York and Los Angeles, mostly, when they're not visiting their parents or walking on the beach in Turks and Caicos. If you, too, live in NY or LA, you will up your chances by 1,000,000 percent of seeing Elizabeth Banks going into a Starbucks.
2. Get a job in a coat check. If you want to actually interact with famous people, get a job working at a place where they go. Everyone needs to eat, right? And famous people often end up eating at hot spots because, whether they like it or not, keeping a high profile is part of their job. I had a roommate in New York who was a coat check girl/cocktail waitress, which is how she met Bono and Nicolas Cage. If being a cocktail waitress isn't your style and you want to meet low-key celebrities, get a job in a famous bookstore. Working the bag check at Shakespeare & Co. is how I met John Malkovich, Michelle Pfeiffer, Stephen Fry, Jodie Foster, and John F. Kennedy, Jr.
3. Live where famous people go to get away. If you don't want to live in New York or Los Angeles, you still have plenty of choices. Famous people like weekend getaways as much as us regular folks do, so living in a pretty town a friendly distance from a big city can be a good choice. Again, having a job where you work with the public is a big help, but just being out and about a lot will help, too. I now live in a town about 100 miles north of L.A. and I've seen more celebrities than you can shake a stick at. I haven't talked to very many of them, because I'm shy and don't want to bother people, plus there's nothing like having Gary Oldman, John Cleese, or Steven Spielberg 10 feet away to make my palms sweat.
4. Be a good spotter. This, of course, is crucial. Celebrities on the street often work not to be recognized, and cover their heads in baseball caps and dark sunglasses. I'm terrible at recognizing them in this state, but I have a lot of friends who are great at it. Stroll down 5th Avenue in New York or hang around a baggage carousel at LAX any afternoon of the week and you're bound to have walked by 10 different celebrities without even knowing it. This is how (I'm told by friends who have better eyesight than me) I walked right past John Irving, Isabella Rossellini, Woody Allen, Dudley Moore, and Philip Glass.
5. Be lucky. Even if you do none of the above, sometimes you're just lucky. Liv Tyler's ex-husband's parents may happen to live right across the street from you and you're just taking the dog out for a walk when she shows up with the baby (true story).
I realize that none of this advice applies if you're looking to meet one particular person and either make them fall in love with you or force your screenplay into their hands. I'm not quite sure what how to help you with that. Maybe try visualization.
Image via csztova/Flickr