What kind of lens works best for extreme close-up or macro experiments on a cellphone camera?

Because your plan is to mount a supplemental close-up lens, in direct contact with the surface of your camera phone, you believe this added lens must have a very short back focus. This is incorrect thinking. In fact, you can procure most any hand-held magnifier and place it in direct contact with the phone. The air-space between the phone and lens will not have that much influence as to the outcome. A magnifying lens placed before the camera lens, acts almost exactly as reading glasses perform. Best would be a jeweler’s loupe. I suggest you procure a Hasting’s 10X loop. The Hasting’s is a three element color corrected magnifier favored by jewelers, stamp, and coin collectors. This would be your best bet.

The 10x or 10 power translates to the distance eye-to-object or in this case, camera-to-object. A 10X magnifier shortens this distance by a factor of 10. The normal comfortable close viewing distance is about 10 inches. Mount a 10x and this distance shortens to 1 inch; hence the object is seen as 10x larger as compared to the unaided eye (camera).