This is a fun topic. Personally, I get so much more excited about seeing someone well-known, but not necessarily famous, who inspires or impresses me on an artistic or personal basis. Way more so than a chance encounter with Paris Hilton, Brangelina, or Tom Hanks (all of which have happened, but left little impression on me).
Just being in the same room as Steven Soderbergh was awesome for obvious reasons, same with meeting Paul Greengrass and Michael Mann. And a couple of crazy nights with John C Reilly and Diego Luna were cool as well.
My uncle has one of my favorite celeb stories though for sure, during which he spent about an hour talking to a guy at a bar before being told it was Billy Joel, this long after the man had reached stardom.