Close to Mae Sai (about 10km outside) is the Fish and Monkey Cave.
You have to turn off the main road. (Coming from Mae Sai to the right).
This is not such a touristy place.
They have a Wat close to a small mountain. Monkeys (Makaken) climb over the temple and the surrounding trees and chase each other.
The Fish cave is a lake that ends directly at the mountain - or better in it, since the cave goes on under water.
In the lake a lot of Koi (special fish - ask any collector how pricey they are) swim.
You can feed the fish and enjoy the surrounding a little.
I really liked the monks around here. Wonderful contrast between their orange robes and the green surrounding.
There are more caves and temples here. Some stairs go up to the Monkey and Lion caves.