Dental Battery
You mention triceratops teeth. well the "teeth" you refer to are not a single tooth but a dental battery, which is hundreds of tiny interlocking teeth. they continuously grow new teeth that get pushed out but due to their shape and being cemented together by tooth formation they stay interlocked with their neighboring teeth creating a large solid chewing/cutting surface. It acts like one huge tooth but is made of many tiny teeth locked together.
As the animal gets bigger the battery gets bigger as the individual tiny teeth get bigger and/or new rows are added, but the process of growth is so gradual compared to how fast individual tiny teeth are replaced the change in size does not disrupt the battery.
Dental batteries are common in ornithischian dinosaurs. And quite frankly is a batter system in many ways than what mammals got stuck with.
An extracted dental battery
A thin section cross section.
https://www.nature.com/articles/s41598-017-16056-3
I want to address a few things you seem to misunderstand.
The teeth have to have a genetically inherited and consistent shape. In other words, if the tooth in one specimen is triangular with ridges along the edge, then that trait should be seen across all members of it's species.*
That is the norm for teeth, there are almost no exceptions to it. Mammals can often be identified by teeth alone.
The tooth needs to have a similar rigidity and sharpness to most earth animals. They cannot break constantly or deform under regular use.
so normal teeth
The tooth has to be vascularized and innervated in some way; so something similar to a root might be necessary.
Normal for amniotes at least. teeth in use are usually vascularized, in tetrapod's the root of a tooth is only absorbed right before it is shed.
The tooth cannot have a 'cap' like seen in the horns of earth animals.
there are no teeth like this, so again normal teeth.
The tooth should be made out of materials found readily in an earth-like ecosystem.
so again normal teeth
The tooth should not dehydrate easily.
this is only an issue in animals that have teeth exposed outside the body, which is only crocodiles, saber tooth cats, naked mole rats, and few other things.
The tooth MUST grow with the animal; keeping the skull it's newborn size is not an option.
This is the hard part. There is only one a way a tooth in use can grow and that is the way rodents and horses do it, by having ever growing teeth. But given your comments elsewhere I wonder if you actually need this, you seem to be confused about how teeth work in most animals. It would help if you clarified why you need this. Normal archosaur teeth may work for what you need, they just grow new teeth in staggered pattern so there is always fresh and worn teeth side by side. before a tooth even gets loose there is already a replacement underneath it and a functional tooth on either side of it. They are never without teeth.
