I am looking for an app that plays the sound of rain, preferably something with for free (although I'd get a paid one if it's worth it).

Ideally, I'd like it to have different levels of rain (i.e. light rain, rain storm, thunder, etc) and something that isn't looped (or at least noticeably looped).

Any recommendations?

Relax Melodies Oriental is a cool app (its icon shows a pagoda in silhouette). I think it was free when I got it, and I learned about it on a tech blog post with app recommendations. It lets you layer different sounds, including different rain sounds, and has a timer you can set if you like.