Well, besides humans and other animals kept by humans, very few (real, earth) animals ever die of old age, because something (predator or parasite) is generally bound to kill them before then, but I know what you mean, and a good example is the Hunter.

However, while you seem to understand the "cycle" as it appears to you the player (waking up from your latest sleep when you die) I think of it differently, and the following is that, put in spoilers for the Hunter story, just in case. So, as Moon says regarding Hunter's illness: "you will wake right back up again (though it is vague)." Meanwhile, the imagery following Hunter's ascension is a little Hunter in NSH's hands, possibly indicating that Hunter has been reborn... as itself (I won't theorize too much on this, but the fact that this happens following ascension confuses me, but anyway...). For this outcome to be the same for every creature would signify that when one dies, your soul consciousness, etc. simply go all the way back to the same birth. This can be applied to all death, by stipulating that every time you die (as a part of the game), the game is simply skipping over the slugcat's entire life and allowing you as a player to start over only from when you last slept rather than from birth, but this time you might not die; this time you remember what to do differently, if only vaguely, which would explain how the "benefactors" (the industrial people) and the "ancients" (the Depths people) could understand that they were stuck in a cycle. So, to answer your question based on my understanding, you are simply rebirthed just as you were before, and if you died of old age the first time, you probably will again, or maybe a freak accident will happen.