Great ones, but late ones unfortunately.
Modern Mathematicians Everyone Should Know

With two mathematicians on the list I have claim to have met (even if communication was limited), I would like to add another who I can say I have known and talked to, John Horton Conway. A real inspiration, and the originator of some wonderful things (and some pretty major maths, and at least one puzzle with a bottle of Jack Daniels as a prize that led to Fields medal.)
Way out of his many pure mathematical fields, he proved that the principles of physics imply that, with suitable definitions "if experimenters have free will, then so do elementary particles."
Still active, despite being 2 days older than my father.

As well as crossing Stephen Hawking's path in the applied maths common room on my very rare visits (I used to jokingly refer to the place with diabolical metaphors) I recall often seeing him buzzing down Mill Lane on his electric wheelchair, leaning his head to the side (which must have made his view ahead slightly strange). Almost 30 years ago now, which is quite extraordinary.

I like a bit of name-dropping.
Checking my facts, I found that while Conway did call the weird connection that led to Fields medal winning work "monstrous moonshine" it was another who offered the prize. But there are many other highly original and quite accessible things due to him to be found in his wikipedia page and elsewhere.
Benard Riemann, student of Carl Gauss.
Srinivasa Ramanujan