Chimney cleanout full of water. Basement leaking
I added a chimney cap, and the leaking was significantly reduced during rains. 


(Image from http://Lowes.com)

I later noticed that during rain, water was pooling on the ground near the chimney. So I built up soil against the base of the house. This made the rain water run down and away. 

After both fixes, there is no longer any water leaking in, even during heavy rains!