Q: Is it possible to catch Mew in Pokemon Leaf Green?
A: Unfortunately there is no way to catch Mew in Pokemon Leaf Green. There was a rumor going around for a while that you could catch him after beating the Elite Four exactly 12 times, doing a couple missions, catching MewTwo, and then going back to where you caught MewTwo and he will eventually appear. Way too many people have tried this though and sat there for countless hours without even seeing him.
Mew is built into the game, but he just isn’t catchable via the normal method. You have to either trade him, or attend a special Nintendo event to get him, but Leaf Green is old now, and they only do events for Diamond/Pearl, so sorry guys.