A simple stack of bricks could go to about 340m before the lowest are in danger of compression failure, based on compressive strengths given below, unchecked. As stated below, the mortar is generally stronger than the brick.
This presupposes that the foundation is inflexible - any movement will put all or part of some of the bricks into tension and you will have cracking almost immediately.
It would be very difficult to construct a brick building more than five conventional storeys high without internal structural bracing. Conventional houses have floors which are laid onto joist hangers; once the structure goes above three floors, there is a requirement for internal bracing so some of the floor joists are structural to prevent rotation of the structure or flexing. Small areas in plan, such as chimneys, can go much higher before this becomes a problem.
Openings for windows and doors present opportunities for the structure to flex as well; this is again where bricks and/or mortar joints are placed in tension with reliably catastrophic eventual results.