Turtle

Turtles are one of the entries people finish a whole world without ever ticking off. They want surface water a couple of tiles deep with grass growing on the floor of it, and plenty of spawn points simply never generate anything that fits. If your world has no lake near home, dig a shallow pond beside a town, let grass creep across the bottom, and check it in daylight.

The statue will produce them on demand, but statue critters puff into smoke the instant you swing a net at them. That makes the statue useless for both the bestiary tick and the cage, so the pond is the only real answer.