First, we need to find out how much space the small bin has. We know that the small bin can hold three-fourths of what the big bin can hold. The big bin has a capacity of 44 towels. So we calculate how many towels the small bin can hold like this:
\[ \text{Capacity of the small bin} = \frac{3}{4} \times 44 = \frac{132}{4} = 33 \text{ towels} \]
Next, we see that there are already 17 towels in the small bin. To find out how many more towels we can fit in, we subtract the number of towels already in the bin from its total capacity:
\[ \text{Remaining capacity} = \text{Capacity of the small bin} - \text{Towels already in the small bin} = 33 - 17 = 16 \text{ towels} \]
So, the small bin can hold 16 more towels!
Therefore, the number of extra towels that the small bin can store is
\[ \boxed{16}. \]