Johns Hopkins gives extremely generous need-based (not merit-based) aid. They calculate COA - EFC - $20.5k unit loan, and the remainder is a need-based scholarship. Since the non-tuition aspects of the cost of attendance (budget for housing/food/etc.) exceed $20.5k, someone with little-to-no expected family contribution could theoretically receive a need-based scholarship that exceeds the cost of tuition. (Another way of putting this is that if every medical student's family contributed as much as their EFC was calculated to be, then no medical student would need to take out more than $20.5k in loans per year.) Hopkins is also generous in its calculation of EFC; my EFC as calculated by Hopkins was lower than at the other institutions I was accepted to. They heavily factor if you have any sibling(s) in college for whom your family is also paying tuition, for example.
University of Maryland SOM offers full-tuition merit-based scholarships to competitive applicants (anecdotally, 95th+ MCAT percentile).