The problem asks us to assign a precise meaning to the expression.
Let $a_0=1$, and for every $n\ge 0$, let
$$a_{n+1}=\sqrt{1+a_n}.$$
The precise meaning of the expression is
$$\rho=\lim_{n\to\infty}a_n.$$
Remark: The limit exists, and a version of your argument shows that the limit is indeed $\frac{1+\sqrt{5}}{2}$.
Here is another example of a similar problem. Assign a precise meaning to
$$\rho=1+2+4+8+\cdots.$$
We could (?) say $\rho=1+2\rho$ and therefore (??) $\rho=-1$. It is fairly unlikely (though not impossible) that we would really want to say that $1+2+4+\cdots$ means $-1$.