What kind of cooler do I need for a 5-day trip without resupply?

A rotomolded cooler like a Yeti or RTIC is the only real option. Cheap Coleman or Igloo coolers only keep ice for about two days in summer. Rotomolded coolers have thick, insulated walls that hold ice for days—even a week—if pre-chilled and kept shaded. You don’t need the most expensive brand; RTIC and similar knockoffs work nearly as well for less.