The ISS has had crew since the year 2000. Do the health issues with microgravity keep compounding, or do they stabilize after a few years and maybe even diminish? I suppose the animal needs to be a mammal, fairly big and long lived.
437 days in a single stretch.
The mammal was a relatively standard specimen of Homo sapiens sapiens. Known affectionately as Valeri Polyakov by his handlers, the subject was launched to orbit on-board Soyuz TM-6 on August 29, 1988.
The mammalian subject was allowed free movement in a spacious 350 cubic meter enclosure, provided with a diet consisting of around 100 g of protein, 130 g of fat and 330 g of carbohydrates per day source, and allowed several trips outside during the course of his mission.
The subject was returned to Earth on-board Soyuz TM-7 on April 27, 1989.
The record for the longest total time any mammal has spent in microgravity is also held by a specimen of Homo sapiens sapiens; Sergei Krikalev, with 803 days in space.
As to the second part of your question:
Do the health issues with microgravity keep compounding, or do they stabilize after a few years and maybe even diminish?
We simply don't know. Even Polyakov's record holding flight didn't break the 18 month mark. The ongoing One Year Mission is a start towards the kind of long duration experiments that will eventually answer your question.