Answer: 502

Small Hint:

Once a set of nonzero digits is chosen, their order is forced

Big Hint:

Exclude subsets of sizes $0$ and $1$ from the subsets of $\{1,2,\ldots,9\}$

Solution:

The digit $0$ cannot occur, because it would have to be the first digit and leading zeroes are not part of a decimal representation. Every subset of at least two digits from $\{1,\ldots,9\}$ gives exactly one ascending integer when written in increasing order. Hence the number is $$\begin{aligned}2^9-\binom90-\binom91&=512-1-9\\&=502.\end{aligned}$$