Up to what thrust setting of a gas turbine engine can the airplane stay stationary on ground?
There are two answers to this question depending on how you look at it. 

If the brakes are on: Full Thrust wont even move the plane this is a requirement of the braking system. 

Brakes off: Somewhere a bit above idle. John covers it in his answer nicely. It will vary by loading and aircraft. 

A propeller plane can get moving even at idle thrust. While jets don't do run ups (unless required for maintenance), propeller planes preform a run-up prior to takeoff where the brakes are utilized to hold the plane in place at a high thrust. A proper short field take off will also involve the application of brakes and full thrust.