It follows by the theory of linear form in logarithms that "few" integers can be written as $2^a-3^b$, with integers $a,b\ge 0$ (see here). What about the case of more than two variables?
Question. Is it true that every sufficiently large integer can be expressed as $$ \alpha 2^a+\beta 3^b +\gamma 5^c, $$ for some integer $a,b,c \ge 0$ and $\alpha,\beta,\gamma \in \{-1,1\}$?
We can definitively say that there's no range of numbers above which "every number is expressible in this form" because there are certain modulus classes you can't ever hit with our sum of three numbers.
In particular, if we are thinking mod $120$, then
$$\pm2^a \in \{ 1, 2, 4, 8, 16, 32, 56, 64, 88, 104, 112, 116, 118, 119 \} $$ $$\pm3^b \in \{ 1, 3, 9, 27, 39, 81, 93, 111, 117, 119 \} $$ $$\pm5^c \in \{ 1, 5, 25, 95, 115, 119 \} $$
The unhittable classes are as follows:
$$\{19, 47, 49, 59, 61, 71, 73, 101\}$$