winch line will be the most firm rope youll be able to buy. it may stretch a bit but it doesnt store that energy. its just tightening the strands in the rope to a certain point. happens every time you release load and handle the rope but not if you load it, release the load, then reload it without messing with the rope to loosen the strands again.
next would be a double braid rope like stable braid or sta-set.
if you want a bit of stretch but not near as much as your old rope gave you, look at the dynamic ropes like Yale dynasorb.
For the record, even cable and chain stretch... why do you suppose they are so dangerous when they break