Passengers left after three quarter-departure stops

A ferry starts with some number of passengers. At each of three stops, three quarters of those on board disembark and then $7$ new passengers board. Passenger counts must be whole numbers at every stage. What is the smallest number that could be on board after the third stop?

$10$. Starting with $x$, after three stops there are $\tfrac{x}{64}+\tfrac{147}{16}=\tfrac{x+588}{64}$ people. The smallest $x$ making this (and every intermediate count) a whole number is $x=52$, giving $52\to20\to12\to10$.