On a 3 by 3 grid, a stack of cubes looks like the left drawing from the front and like the right drawing from the right-hand side. What is the greatest number of cubes it can contain?

Each column can be no taller than the front view allows for its line and the side view allows for its row, so the most is the smaller of the two in every square. Rows (front to back) allow 2, 3 and 1; lines (left to right) allow 3, 3 and 1. Left line: $2+3+1=6$; middle line: 6; right line: $1+1+1=3$. Greatest $6+6+3=15$.