If hiking by myself I carry a small first aid kit - bandaids, gloves, cleaner, and a ankle wrap. If hiking with a larger group and/or overnight, I'll carry a larger first aid kit with non-Rx medications, more gloves, sam-split, larger bandages, paramedic shears, tape, iodine, anti-bacterial gel, CPR mask, etc.
Maybe, just maybe, the best thing one can carry with them are common sense and compassion. It's a very challenging environment when one is 5-10 miles from the road and certainly more than a hour from an ER. Using what one has on hand and keeping in mind doing-no-harm and helping another are probably the best one can do. One can improvise in the wilderness.
Training helps dearly - I don't intend in the least way to knock that - I well understand the value of education. CPR and first aid are good things to know. I've had some minimal first responder level training and seminar courses, but I don't currently run with an ambulance or fire service, so I haven't gone for an EMT cert, etc. I even used to work as an engineer on heart monitors. But I'll be the first to admit that it's really easy to forget things, without constant use of those skills, no matter what the level. Right now, renewing my certs every 2 years is about it.
BTW, my hat's off to anyone who does maintain EMT, EMT-I, EMT-P, etc level skills or higher (nurses, PA's and Physicians). It's very hard work and the pay off is in knowing more about how to handle and treat emergency medical situations, and in helping others. It's a great thing.
--LTH