How is it possible that multiple overtones can exist at the same time?

Your mistake is in your initial assumption:

When one pulls a string, it starts to oscillate and forms a standing wave with frequency $$f=f_0$$

The only way this is true is if you could start the string with length $L$ and wave speed $v$ off with the exact shape of the sine wave of your fundamental frequency: $$y(x,t=0)=A\sin\left(\frac{\pi}{L}x\right)=A\sin\left(\frac{2\pi f_0}{v}x\right)$$

Of course, when you pluck a string you are not starting the wave off with this shape. You are approximately starting with a piece-wise function: $$ y(x,t=0) = \begin{cases} \frac{2A}{L}x, & 0\leq x\leq L/2 \\ 2A-\frac{2A}{L}x, & L/2\leq x\leq L \end{cases}$$

This initial function can be expressed as a linear superposition of your overtones: $$y(x,t=0)=\sum_{i=1}^\infty a_n\sin\left(\frac{2\pi nf_0}{v}x\right)$$ and it is the values $a_n$ that determine "how much" of overtone $n$ is in your wave. If you find values for each $a_n$, you will see that $a_1$ has the largest value, since our initial waveform is somewhat shaped like the waveform of the fundamental frequency. If you were to pluck at other locations or in other weird ways you could excite other overtones more than the fundamental frequency.

If there are overtones, then the "wave" is not just of the main frequency, but is more complicated. In the linear approximation the tone and overtones may come together - in a superposition, due to linearity of the wave equation (which admits very complicated "wave" profiles).

You may excite any overtone without exciting the fundamental tone, so it is not always that the main tone has the highest amplitude.