you could literally get to puerto natales with nothing and outfit yourself for the trek. I wouldn't recommend it unless you like to carry lots of things strapped to your pack, but it certainly could be done.
There is an issue of food – Chile does NOT want you to bring food into the country. The website says if it is PROCESSED food (i.e. Mountain House) in its original packaging it's OK, but nothing fresh, no nuts, etc.
So I brought some MH meals and repackaged them at the hostel before we left. Then went grocery shopping and bought lots of other stuff to round out my food supply. At the time you couldn't really buy any food on the trek unless you are staying at a refugio, but it sounds like that may be different now.
Do NOT expect a wilderness trek – it is decidedly NOT that. Very crowded (not nearly as much on the back side tho!) but gorgeous. You can read my trip report at backpackerPT.com if that interests you.
We stayed at the Singing Lamb hostel in PN and we LOVED it. Quiet, great food, nice operators.