Perpetual stew

A perpetual stew, also known as forever soup, hunter's pot, or hunter's stew, is a pot into which foodstuffs are placed and cooked, continuously. The pot is never or rarely emptied all the way, and ingredients and liquid are replenished as necessary. Such foods can continue cooking for decades or longer if properly maintained. The concept is often a common element in descriptions of medieval inns. Foods prepared in a perpetual stew have been described as being flavorful due to the manner in which the ingredients blend together. Various ingredients can be used in a perpetual stew such as root vegetables, tubers (potatoes, yams, etc.), and various meats.

Historical examples
Perpetual stews are speculated to have been common in medieval cuisine, often as pottage or pot-au-feu:

"Bread, water or ale, and a companaticum ('that which goes with the bread') from the cauldron, the original stockpot or pot-au-feu that provided an ever-changing broth enriched daily with whatever was available. The cauldron was rarely emptied out except in preparation for the meatless weeks of Lent, so that while a hare, hen or pigeon would give it a fine, meaty flavour, the taste of salted pork or cabbage would linger for days, even weeks."

A batch of pot-au-feu was claimed by one writer to be maintained as a perpetual stew in Perpignan from the 15th century until World War II, when it ran out of ingredients to keep the stew going due to the German occupation.

Modern examples
The tradition of perpetual stew remains prevalent in South and East Asian countries. Notable examples include beef and goat noodle soup served by Wattana Panich in Bangkok, Thailand, which has been cooking for over 49 years, and oden broth from Otafuku in Asakusa, Japan, which has served the same broth daily since 1945.

Between August 2014 and April 2015, a New York restaurant served a master stock in the style of a perpetual stew for over eight months.

In July 2023, a "Perpetual Stew Club" organized by social media personality Annie Rauwerda gained headlines for holding weekly gatherings in Bushwick, Brooklyn, to consume perpetual stew. Hundreds attended the event and brought their own ingredients to contribute to the stew. The stew lasted for 60 days.

Mathematical model of average age
Let $D\in\mathbb{N}$ be the time since the stew's creation in days, and let $$p\in[0,1)$$ the percentage (where $$p=0.17$$ equals to 17 %) of stew left uneaten in the pot after every day. The stew is filled with fresh ingredients and stirred thoroughly at the beginning of each day. Then the average age (in days) of the ingredients within the stew at the time of the refilling is given by the partial sum  $$A_{p,D}=\sum_{d=0}^D d\,p^d=\frac {D\,p^{D+2} - (D+1)\,p^{D+1} + p}{(1-p)^2},$$whose age limit $A_p$  with respect to $D$  is given as the series$$A_p=\lim_{D\to\infty} A_{p,D}=\sum_{d=0}^\infty d\,p^d=\frac {p}{(1-p)^2}.$$The partial sum $A_{p,D}$  consists of nonnegative summands, hence increases as $D$  or $p$  is increasing, and $A_p$  is the limit and an upper bound for $D\to\infty$, hence $A_{p,D}\leq A_{p}$ . The limit $\lim_{p\to1} A_{p}=\infty$  tends to infinity and behaves as naively expected. Naturally, $A_{p,D}$ and $A_{p}$  describe upper bounds if $$p\in[0,1)$$ is an upper bound for the amount of stew left in the pot after every day. Trivially the continuation is $A_{1,D}=D $  and $A_{0,D}=0 $.