The One Ring may exist in a low-magic setting, but it was still unfathomably powerful, as it was a relic of an older time when magic was still strong in Middle-Earth. Gandalf and Galadriel, despite being so powerful they could reshape reality with their words alone, were wary of the One Ring. The mere thought of possessing it corrupted Saruman. Due to the fact that Sauron first gave the Elves the secret of crafting the Rings of Power, the One Ring's existence kept them going- when the Ring was destroyed, all works created by the other Rings of Power would slowly fade. Almost all magic in Middle-Earth at the time of the books depended on the One Ring, and with its destruction magic began to leave Middle Earth. This is all ignoring the fact that only Sauron could use the One Ring to its fullest potential.
Admittedly, Sauron played second fiddle to quite a few powers in Middle-Earth. The Numenoreans, for example, absolutely humiliated him during the Second Age. In their arrogance, they took Sauron alive, where he was able to corrupt them into going to war with God, who smote Numenor. One of the great themes of Tolkien's work is that power corrupts, and absolute power corrupts absolutely.
I guess what I'm getting at is if Fat Rick had access to the One Ring, he'd turn into a fatter version of Gollum.
