4/9 divided by 2/3 divided by 5/6

Depends. Division is not associative.

(4/9)÷((2/3)÷(5/6)) = (4/9)÷(2/3 * 6/5)
= (4/9)÷(12/15) = (4/9)÷(4/5) = 5/9

((4/9)÷(2/3))÷(5/6) = (4/9 * 3/2)÷(5/6)
= (12/18)÷(5/6) = (4/6)÷(5/6) = 4/5