Some of the franchised gyms do this but in my experience local gyms often do not. At my local gym their memberships are for a "defined term" (3 months, 6 months, etc.) and if you don't renew, they end. I've never tried to end one early but knowing the owner and how he runs the place I am quite sure it would not be an issue.

You can also just pay as you go, i.e. per visit but that ends up being a lot more expensive.