Given $f(x) = a \sin^3 x + b \tan x + 1$, if $f(2) = 3$, then

$$f(2\pi - 2) = a \sin^3 (2\pi - 2) + b \tan (2\pi - 2) + 1$$
$$= -a \sin^3 2 - b \tan 2 + 1$$
$$= - (a \sin^3 2 + b \tan 2 + 1) + 2$$
$$= -3 + 2 = -1$$

Therefore, the answer is $\boxed{-1}$.

This problem can be solved by using trigonometric identities and the values of trigonometric functions, as well as applying the properties of odd and even functions, testing computational skills.