It depends.  According to the IRS, depending on what the concierge fee is specifically for, some, all, or none of it may actually qualify as a medical expense.

If the monthly fee merely grants you access for future services, with no services actually received, it is not deductible. 

If the fee includes any medical services, it may be deductible depending if the included items would meet the IRS definition of  a deductible medical expense.  (ie. If the fee includes an annual physical, it is deductible because an annual physical is a deductible expense.  If the fee allows for free OTC medications, it is not deductible because OTC medications are not deductible.)

Thank you for your answer.  How much of the fee would be deductible? Would the deductible amount be limited to the deductible expense, or would the entire fee be deductible?

This is somewhat uncharted territory for the IRS, so I would be conservative in my deduction.  If your monthly payment is $1,000 and in return you get two free office visits per year, I would deduct the value of the free office visits rather than the $12,000 total.  If your monthly fee is somewhat in line with the services you receive in return, I would be okay deducting the full amount.  I've seen a lot of discussion on this, and the word "reasonable" is often used by the IRS.