Rather than pitch in myself (as I’m not knowledgeable enough to offer an opinion) can I point the good dopers in the direction ofthese podcasts.
I’m pretty sure that our overseas friends are able to listen to and download the radio programs, apologies if not.
Ten, 15 min potted histories on the great mathematicians by the BBC and presented by Marcus Du Sautoy. I’m sure it isn’t exhaustive but they certainly give the uninitiated a neat overview of who the main players are and what they did.
I don’t understand math well. However I’d guess Von Neumann, Newton and maybe Tao.
Most of what we think of as Calculus predates Newton. Even the Fundamental Theorem more or less pre-dates him (and Leibnitz), though he systematized its use.
His main skill was applying math to physics problems, so I agree with the above posters that he’s a strange choice for Best Mathematician, though he’d certainly be in the running for Best Physicist.
I’d go with Gauss, Euler and Archimedes.
If you’d like to look at a book that gives you a pretty comprehensive list of the most important mathematical breakthroughs and thus indirectly the most important mathematicians of all time, get hold of Maths in 100 Key Breakthroughs by Richard Elwes. It covers discoveries from prehistoric times to this century. I think it’s a lot harder than you may suspect to compare the mathematical abilities of mathematicians from different periods of history.
Gauss and Euler are musts on a list of greatest mathematicians. The third I am more ambivalent on. Archimedes is a good choice. As is Newton or Leibnitz. Fermat would probably be my third, if you put a gun to my head. But also Euclid. Ugh.
I’d put Fermat last. 
Seriously, though, I think Archimedes has to be there just because of what he did when.
In no particular order…
[ul]
[li]My 8th grade Algebra teacher[/li]
[li]My Calc 1 professor[/li]
[li]Donald Duck (in Mathmagicland)[/li][/ul]
My list would probably go Euclid (for lasting impact), Gauss, and Euler.
Lots of honorable mentions, though, including Lobachevsky, Erdos, Pascal, Hilbert, Russell, Chebyshev, and a bunch of others.
Urban legend. He would have been college educated, except he would only study math and ignore everything else. He did certainly do a lot of original research on his own, though.
Euler, Fermat, Gauss (EGG for short)
The guy who first figured out that 1 + 1 = 2
The difference between the greatest and the top three greatest is clearly a math question, so it fits right in.
Einstein wasn’t a mathematician.
That’s two people, Alfred Whitehead and Bertrand Russell. After twenty years they figured out that they couldn’t prove it.
I’ll go with Archimedes as number one, and the other two would be the proverbial constipated mathematicians who worked it out with pencil and paper.
It’s hard to judge how to rank Euclid, because we don’t know how much, if any, of his Elements is original to him.
I’ve heard “Archimedes, Newton, Gauss” as the “canonical” answer to the thread’s question (maybe given by E.T. Bell?), but IMHO Euler deserves to be in there, maybe replacing Newton for the reasons others have mentioned.