It seems that the last power of $2$ that only has $1$'s and $2$'s in its ternary expansion is $2^{15} = 1122221122_3$. Empirically, this is true upto $2^{10^7}$. Is it true in general?
The context of this question is that some of us were looking at the operation of given a number taking the product of its digits. The base $10$ case seemed too complicated, and the first non-trivial case is base $3$, where every number immediately reduces to a power of $2$ (or $0$).