The Cavern of Sleep is filled by a wide variety of formations including stalactites, helictites, beaded helictites, anthodites, and soda straws among others.
Because the Timpanogos cave system is still a wet cave system we know that the cave formations are still developing or "growing." Some people wonder if the whole cave will one day fill up with formations making a giant block of calcite. While theoretically this is possible, it is highly improbable that the caves will fill up with formations.
There are two main reasons that this is unlikely to occur. One factor is the location of the caves in what is considered to be an active fault zone. Cave formations develop so slowly that a major earthquake will probably strike the Timpanogos cave system destroying the caves before they fill up. The second and more recognizable threat to the caves is that of erosion and particularly mass wasting in the American Fork Canyon. These same forces which exposed the current cave entrances will likely open up bigger and bigger entrances until the caves dry out and stop growing or eventually tumble down the American Fork Canyon themselves.
Don't worry too much about these possibilities though; it may be thousands of years before any major geologic events affect the caves. The important thing to remember is to enjoy and protect the caves now so that as many people as possible will be able to share them. Also, on the bright side, erosion may expose entrances to new and wonderful caves that have not yet been discovered!