The front view, the side view and the plan from above of a stack of cubes on a 3 by 3 grid are drawn. What is the greatest number of cubes the stack can have?

Only the 6 shaded squares of the plan hold columns. Each column can be as tall as both its line in the front view and its row in the side view allow. Back row (allowed 2): 2, 2, 2. Middle row (allowed 1): its one column 1. Front row (allowed 3): left line 3, middle line 2. Total $6+1+5=12$, and this stack does show the drawn views.