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John Baez is a mathematical physicist at the University of California at Riverside. His "fun stuff" page displays his amazingly wide interests. He is a cousin of the singer, Joan Baez.
Timothy Gowers, Rouse Ball Professor at Cambridge University, won the Fields medal in 1998 (find out why here), but has found time to construct a web page that is well worth browsing. He also has a mathematical discussions page, containing "ideas that I have come across in one way or another and wish I had been told as an undergraduate."
Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University is not the answer to the question "Which living mathematician has published the most papers?", but has published over 1,100 articles.
Neil Sloane's web page is a treasure trove for information about discrete mathematics: How best to pack balls, how best to distribute a finite number of points evenly on a sphere, and more. For example, if the finite sequence 1, 2, 3, 6, 11, 23, 47, 106, 235 crops up in your work, the On-Line Encyclopedia of Sequences will offer some of the nearly 100,000 infinite sequences in its database that begin with these integers. There is also a link here to the intriguing Inverse Symbolic Calculator - enter a decimal number and get a list of possible formulas that agree with it up to the given decimal places!
Terence Tao, Professor at UCLA and a 2006 Fields medal recipient, has mathematical interests ranging from PDE's to number theory. On his web site you'll find links to most of the really useful mathematical sites, and much, much more.
"Which prominent mathematicians have blogs?"
Terence Tao's blog.
Gil Kalai's blog.
Tim Gowers's blog.
Links to many other mathematical blogs, wikis, online encyclopedias, etc. can be found here.
The general fascination with Erdös Numbers is a bit mysterious to me. If they interest you, the Erdös Number Project site will keep you occupied for hours. Actually, there are some rather interesting statistics and trivia to be found here, for example: The fraction of mathematical papers authored by just one person has steadily decreased over time, starting out above 90% in the 1940s and currently hovering just under 50%.
Until recently, the answer to the question "Who is the oldest living mathematician?" was Leopold Vietoris, who died on April 9, 2002, at the age of 110, and was also the oldest living Austrian. He may be the only mathematician on the oldest human beings list.
Carl Friedrich Gauss has been called the "Prince of mathematicians." But who is the "Prince of living mathematicians"? Some might say Jean-Pierre Serre, winner of the 2003 and first Abel Prize. Another candidate until his death on 13 November, 2014 was Alexander Grothendieck, one of the most intriguing personalities in modern mathematics. The (incomplete) translation into English of his book of reflections Récoltes et Semailles ("Harvests and Sowings") offers a glimpse into this extraordinary mind.
Where do mathematicians most like to meet? The Oberwolfach Mathematics Institute in the German Black Forest, perhaps. Here there are 51 week-long meetings per year, each focussing on an area of mathematics.
Who's the most opinionated mathematician? Well, Doron Zeilberger's opinion on colloquium etiquette is number 59 on his list!
Who are your mathematical ancestors? Find out (possibly, since the site's database is certainly not complete) at the Mathematics Genealogy Project site.
You may have noticed that this is a quote-free site. If you're disappointed, help yourself at the Mathematical Quotations Server.