Having just attempted a no skill points run, I believe you need Hammer Throw by stage 11, but can finish the game without further upgrades. If you want to collect all the exp as well, Fire Arrows are necessary by stage 10. I didn't skill Fire Arrows until I'd beaten Stage 12, so they're definitely not needed to win the game.
For the first instance of why each of those upgrades are necessary:
Fire Arrows: On stage 10, there is a part fairly early in the level where there is a wooden box containing an Experience Bottle atop a switch which keeps a stone door shut, locking away access to a Large Experience. While it's actually possible to conjure a wizard box and get things positioned just right to get the Large Experience, the wooden box must be broken in order to be able to slide its bottle out to yourself through the narrow gap. The gap seems too narrow to get a Thrown Hammer through, but a Fire Arrow can break it.
Hammer Throw: On stage 11, there is a section with a lever that starts out doing nothing, because air needs to be connected in the room above it. Blocking entrance to the room with the air is a stone wall, too high to be reached with your basic abilities. Also, directly to the left of the lever is an invisible wall which prevents you from bringing objects from earlier in the level. You don't actually have to reconnect the air, but behind the rock wall is the barrel that you need for the nearby water and, due to the invisible anti-object wall, one from earlier in the level cannot substitute for it.
As evidence, I present my Let's Play of the game. (Last few videos will be up over the next couple weeks)