Alexander Hamilton
Alexander Hamilton failed to graduate from Columbia's predecessor, King's College, over 220 years ago. (Really, it wasn't his fault. While many maintain Hamilton was a dropout, he only left because the school closed during the Revolutionary War). He remains the most famous person ever associated with the school. Even though he never got a degree and it wasn't even called Columbia at the time.
Hamilton was originally interested in attending Princeton (then hideously named the "College of New Jersey"), but was rejected. When he became a general during the Revolutionary War, he famously ordered a cannonball fired at Princeton's Nassau Hall when a battle was taking place on the college's campus.
A proud pseudo-alumnus, Hamilton was instrumental, with John Jay, in reinstating King's as Columbia College when the war was over.
Hamilton Hall is named after him, and that statue outside Hamilton, yeah, that's him too. You might have also seen him featured on the $10 bill. Oh, and he was murdered by the sinister Aaron Burr, who happened to be a graduate of darkest Princeton.
