3-4 days is the suggested
today might be iffy. e.coli is a bitch.
you're not going to get ecoli from a 4 day old hamburger any more often than you would from one fresh off the grill
if a trace amount of e.coli is present in the original burger, and that burger has not been thoroughly cooked, the e.coli will continue to multiply, increasing your risk eating the leftovers if they are again undercooked. this is the same for any food containing ground beef. Many,many people do not get their burgers cooked sufficient to kill e.coli. If the meat is not 160 degrees then e.coli can remain, and that is more than simply saying "no pink". And a LOT of homecooked burgers have some pink.
now, most people will get it cooked properly, but there's a risk there. I don't eat leftover hamburgers because IMHO they get too dry when reheated properly.