Baltic Way shortlist

Let there be an operator $\ast$. Given an expression that includes this operator, one can make the following transformations: 1. An expression of the form $x\ast (y\ast z)$ can be rewritten as $((1\ast x)\ast y)\ast z$; 2. An expression of the form $x\ast 1$ can be rewritten as $x$. The transformations may be performed only on the entire expression and not on the subexpressions. For example, $(1\ast 1)\ast (1\ast 1)$ may only be rewritten using the first kind of transformation as $((1\ast (1\ast 1))\ast 1)\ast 1$, but it cannot be transformed into $1\ast (1\ast 1)$ or $(1\ast 1)\ast 1$ using a single step – in the latter two cases the second kind of transformation would have been applied just to the left or right subexpression of the form $1\ast 1$. For which natural numbers $n$ can the expression $1\ast (1\ast (1\ast (\cdots \ast (1\ast 1))))$ be rewritten to an expression that does not include a single occurrence of the $\ast$ operator?