Evidence of life first shows up in the fossil record around 3.5 billion years ago, shortly after the Earth started cooling off. This was a great start, but at that point, all life consisted only of single-celled organisms, too small to see. They would not have made good pets.
What happened next? Very little. Three billion extraordinarily boring years went by before complex animals started to evolve.
Then, quite suddenly, over a period of merely five million years (barely two tenths of one percent of the history of life at that time), crab-like animals evolved that were the size of cats. Unfortunately, they did not make very good pets either. But they were not very boring, at least not when compared to bacteria and algae.
This period of rapid expansion is called the Cambrian Explosion.
Cats are a million times larger than single celled organisms. In the last 540 million years, animals have only evolved to be a few times larger than that. Elephants seem large, but they are only ten times taller than cats.
If animals could evolve to be a million times taller than cats, they would be one hundred miles tall. You only have to go up sixty two miles to get to the edge of outer space, so these super-tall animals would have obvious problems in the breathing department, not to mention finding enough food to eat, freezing to death, or spontaneously collapsing under their own weight. Would animals like these make good pets? Yes, provided that you owned a pooper scooper the size of Manhattan.
So, once multicellular animals started evolving, they quickly reached the practical limits imposed by gravity and common sense, and that has pretty much been it as far as size is concerned for the last half billion years.
The pinnacle of evolution was reached approximately five million years ago, with the appearance of the superior beings that we now know as domestic cats. According to scientists, cats represent the most ideal pet that could theoretically exist, if you ignore the possibility of pets that are made out of chocolate.