Object loops are not infinite due to The Multiverse.You do not actually travel in your own timeframe, you simply go to the next timeframe over while alternate you comes to yours. Say, for instance, you take object X from your desk and place it on your desk yesterday, then really you're just shifting the object X over one Multiverse each "loop". Many, many multiverses over, entropy will kick in and object X will degrade and the corresponding you will decide object X looks shoddy and worn out and you'll decide to take object Y instead. Yes, there's going to be an origination and an end point, but so many universes are in between that the chances of our hero being in either one of those two universes is slim. So while a character in fiction my create what appears to be a stable time loop, it is merely a part of a long progression of loops connecting parallel universes. The object has ceased to move down the timeline vertically with everything else, it's moving horizontally. Eventually the object will degrade and things will be done differently but not in the universe covered by this specific story you are reading/watching/whatever.
There is a beginning to a Stable Time Loop.A Stable Time Loop occurs when an event in the future causes and event to happen in the past. However, because only major/important events are kept in history, certain minor events or properties can change without affecting the overall timeline. This allows time loops to have a beginning that isn't related to a Time Paradox. Take Ocarina of Time's Song of Storms for example. The song is first learned in where the windmill guy in Kakariko teaches Adult Link the song. Adult Link then goes back into the past and, as Young Link, teaches the song to the guy in the windmill, causing the time loop. However, it can be said that this all began when another boy with an ocarina (let's call him Bort) learns the song from the Composer Brothers, then teaches it to the windmill guy - not a paradox. The windmill guy then teaches the song to Adult Link. This causes a timeline break where the kid who teaches the song to the windmill guy changes from Bort to Young Link, beginning the loop and possibly removing Bort from the timeline.