In my setting I have a species of anthropomorphs (basically humans) that grow from baby to full-size in the shortest time period possible. The babies are the size of human babies. How short can that time period get, and what modifications should their body/genetic material have to grow at an incredibly fast pace?
I could name a few factors.
Some species with short lifespans might be forced to procreate at a younger age and spread their genes as quick as possible before getting eaten by predators. Which means that females will be forced to give birth at a much younger age. Over a long period of time natural selection will cause males to pick females who are the most fertile. The females who become fertile the quickest will be able to pick the best genes from all the males.
This will cause gene mutations in both males and females of your species to reach puberty and adulthood at a much faster rate.
Other factors are hormones. Providing higher levels of Human growth hormone , estrogen and testosterone can accelerate the the growth rate of your antropomorphs/humans aswell.
An example of another animal of similar body mass to humans is a cougar (puma, mountain lion). These reach maturity in two or three years.'
HOWEVER the puma's brain development is not comparable to that of humans. More similar animals, like chimpanzees and orangutans, take ten-plus years to mature because, more similarly to humans, their brains require significant maturation after birth. Not to say that a cougar's or wolf's brain doesn't change and mature after birth -- but they are less helpless when born and don't need to reach the level of mental maturity a human or great ape does.
Physical maturity, then, might be possible in less than five years, perhaps as little as three (learning to walk upright is more complex than learning to walk on all fours, for instance, and doing so requires more growth in the legs, spine, and related musculature), but mental maturity (given similar final mental condition) seems unlikely to be completed in less than fifteen to twenty years, as is the case now.
Adult is kind of iffy here, because it is possible to grow to full size significantly before reaching mental maturity. This happens in a lot of animals, though obviously the most familiar example is dogs 🐕.
I am going to assume you meant full-size from context.
Humans grow unusually (and kind of pointlessly) slowly. For comparison a beef cattle 🐄 only live 2-3 years.
Humans would almost certainly reduce full-size time down to 3 years or younger eventually with enough survival, just because it gives a massive inherent advantage in education 🏫 and training, making life easier and increasing effective intelligence. So anthropomorphs could have just been around for a long time.
As for further increases in speed, the easiest way is to just grow cells outside the body and inject them in. This could be their own, or you could make them feeder cells grown. Maybe in a plant-esq thing. This is far quicker than digestion.
As for other measures, maybe having a cartilage proto skeleton before the bone one.
Kind of hard to predict those effects, as no animal has that growth strategy, mostly because civilisation to support it has not been around long enough to develop it.
So, in all, 3 years or lower is easy naturally, if other measures are employed, it will be a lot faster, but very hard to predict a limit.
There is substantial variation within existing humans on the age vs height for children. I have shown only the first graph from here.
Note that the lines on this graph may be misleading. A baby born "on" one of these lines does not necessarily stay there. These are the percentile lines so, by definition, they go up as they do. A child might move from one line to another, growing faster or slower than their cohort.
You could get a substantial start by determining the factors that produce this range. There will very likely be genetic factors. There will also very likely be factors of diet.
So, if there were some factor that required babies to grow quickly, a species might rather quickly evolve to match. Maybe there is a predator that only takes prey under a certain size. Or maybe there is a harsh winter such that smaller people tend to freeze. Those with genetics to grow quickly, and who live in cultures where the right kind of food is provided for them to grow, will suddenly be tall. The ones that don't will tend to die out.