Prompted by seeing Does each student need a pet?
It's stated in the letters to first year students:
Students may also bring an owl OR a cat OR a toad
— Philosopher’s Stone, chapter 5, Diagon Alley
However, Ron is able to bring Scabbers (a rat). Which, is not included on the approved list. So, I was wondering, are there any other students who have pets that aren't included on the approved list, and what are the actual limitations for what pets are accepted and which ones aren't. (ex: could a student bring a small dog instead of a cat? Or a student bring another type of bird other than an owl? Or a type of lizard, turtle, snake, instead of a toad?)
Also, related question: Why was Ron Weasley permitted to bring a rat to Hogwarts?, though that question is asking specifically why Ron can bring Scabbers, not about the rule in general, or if other students have pets not on the approved list.