As in, up to structure types...
For example, in algebra, we start with groups (most basic), then rings, then vector spaces, then inner product spaces, then Normed Linear Spaces, and so on, with the main motivation behind these structures is that the interactions in different structures/frameworks are different and also sometimes useful in other areas of interest.
Up to isomorphisms or structural equivalence ideas, are there infinite or finite number of structures?
It would either seem that either: you are now not sure about the question and it is a bit unclear or alternatively the answer is "obviously there are infinite structures".
I guess that ultimately the question I am getting at is: is it possible for a human to know all the possible structure types(and have some intuition for each one?) Does it end? Is there going to ever be a point where someone finally finishes a book on analysis (for example) and says "This is the end. The theory is complete. There isn't anything else which is useful to consider?"
We are in the realm of either algebra or analysis here (you can think of the two realms separately).
It is also possible that this question makes no sense because my intuition of maths is wrong. However, to me it seems intuitively like a very human and very curious question to ask. If you look at complicated advanced analysis book with loads of symbols... and then an even more complicated, more advanced analysis book, I can't help but wonder if the whole theory ends somewhere...