A real number $r>0$ is written. If there is a number $a$ written, we are allowed to write down $a+1$. If there are numbers $a,b$ written (possibly $a=b$ but they must be written twice in that case), we are allowed to write down the (0, 1, or 2) real roots of $x^2+ax+b$. Is it true that we can always eventually write down the number $\sqrt{r}$?
A particular case where this is certainly possible is if $\sqrt{r}$ is an integer. Starting with $r$, we can write down numbers $2s$ and $s^2$ for some integer $s>0$. The root of $x^2+2sx+s^2$ is $x=-s$, so we can now get all integers at least $-s$. This includes all positive integers.
Given $r>0$, we have $r+1$.
So, we have the roots of $x^2+(r+1)x+r=(x+r)(x+1)$, which are $-r$ and $-1$.
Adding $1$ to $-1$, we have $0$, and thus we have the roots of $x^2+0x-r$, so we have $\sqrt{r}$.