What is the recreational pastime of exploring wild cave systems called in the Americas?

We call it spelunking in NZ too. I can remember wearing a hard hat with a carbide lamp on it, jumping over chasms and walking through an underground river. Glow-worms everywhere.

Although not in the category of wild caves, my family visited the Oregon Caves in the late 60's and it was fascinating.

Rosie-knees, The attraction is the thrill of the unknown & exploration itself; the anticipation of what is next, what will be found.