Any number that can be obtained after a finite number of steps is of the form $r\sqrt{2} + s$, where $r$ is positive and $s$ is a nonnegative rational number. Indeed, the initial number $\sqrt{2}$ is of the form $1\sqrt{2} + 0$, and if we multiply a number of the form $r\sqrt{2} + s$ by a positive rational number $t$ or add a natural number $k$ to it, the result is $rt\sqrt{2} + st$ or $r\sqrt{2} + (s + k)$ correspondingly, which is of the same form.

Show that any number of the form $r\sqrt{2} + s$, where $r$ is positive and $s$ is a nonnegative rational number, can be obtained by at most 3 steps. If $s > 0$ then multiply $\sqrt{2}$ by a positive rational number $\frac{r}{s}$, add 1 to the result, and then multiply the result by the positive rational number $s$. If $s = 0$ then it is enough to multiply $\sqrt{2}$ by the positive rational number $r$.

Finally, we show that 2 steps are not always enough. First, make sure that in the representation $r\sqrt{2} + s$ of a number, $r$ and $s$ are uniquely determined. Indeed, if $r\sqrt{2} + s = r'\sqrt{2} + s'$ then $(r - r')\sqrt{2} = s' - s$. If $r - r' \ne 0$ then $\sqrt{2} = \frac{s' - s}{r - r'}$, which is not possible, because $\frac{s' - s}{r - r'}$ is a rational number, but $\sqrt{2}$ is an irrational number. Hence $r - r' = 0$ or $r' = r$; from $(r - r')\sqrt{2} = s' - s$ also $s' = s$.

Consider the number $r\sqrt{2} + s$ where $r = 1$ and $s$ is an arbitrary fractional number. We cannot get this number by multiplications only, because then $s$ remains zero, or with additions only, because then $s$ would be an integer. If we multiply by a rational number $t$ and add a natural number $k$ in either order then the result is either $t\sqrt{2} + k$ or $t(\sqrt{2} + k)$. In both cases we must have $t = r = 1$. But multiplication by 1 does not change the result, and we already saw that by addition only it is not possible to get the result.