Is it possible to construct a sequence that ends in 1000000000?
We first note that we can attain $1000$ from $1$ by $$1,2,4,8,16,32,64,128,256,512,125,250,500,1000$$ Therefore $1000^n$ can be attained by performing the same operations as above with $1$ replaced by $1000^{n-1}$.
On the other hand, note that rearranging the digits does not change the remainder when divided by $3$, and $2^n \neq 0 \pmod{3}$. Therefore the sequence would not reach any multiple of $3$.