The only professional refutation I was aware of came from the neuroscientists Sam Wang and Sandra Aamodt (and their book Welcome to your brain). Today, I consulted the myth-related Wikipedia entry and found other refutations exist (based on studies of fully functional and dysfunctional brains). What I offer below is my own "proof" that if we really used 10% of our brain potential it would be impossible to reach 100% of it.
Fact 1. The human brain uses 20% of the energy consumed by the human body.
Assumption 1 (i.e. the myth). Humans use only 10% of their brain potential.
Theorem. If Assumption 1 was true, it would be impossible to reach 100% of human brain potential.
"Proof". If 20% of the energy consumed by the human body powers 10% of the brain potential, then in order to reach 100% brain potential the brain has to consume 200% of the energy consumed by the human body, which is absolutely impossible*. QED
*as long as the brain is a small part of the human body and is not an organ shared by several bodies